Nuprl Lemma : rmul-distrib1
∀[x,y,z:ℝ]. ((x * (y + z)) = ((x * y) + (x * z)))
Proof
Definitions occuring in Statement :
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
radd: a + b,
all: ∀x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
top: Top,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rmul: a * b,
reg-seq-mul: reg-seq-mul(x;y),
reg-seq-list-add: reg-seq-list-add(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
int_nzero: ℤ-o,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
req-iff-bdd-diff,
rmul_wf,
radd_wf,
req_witness,
real_wf,
reg-seq-mul_wf,
accelerate-bdd-diff,
less_than_wf,
reg-seq-list-add_wf,
cons_wf,
nil_wf,
length_of_cons_lemma,
length_of_nil_lemma,
nat_plus_wf,
regular-int-seq_wf,
length_wf,
accelerate_wf,
bdd-diff_wf,
squash_wf,
true_wf,
reg-seq-mul-comm,
iff_weakening_equal,
bdd-diff_functionality,
rmul-bdd-diff-reg-seq-mul,
bdd-diff_weakening,
reg-seq-mul_functionality_wrt_bdd-diff,
reg-seq-list-add-as-l_sum,
map_cons_lemma,
map_nil_lemma,
l_sum_cons_lemma,
l_sum_nil_lemma,
bdd-diff-add,
cbv_list_accum_wf,
int-value-type,
false_wf,
le_wf,
mul_cancel_in_le,
absval_wf,
subtract_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
absval_nat_plus,
equal-wf-T-base,
absval_mul,
nat_wf,
all_wf,
nat_properties,
equal-wf-base,
nequal_wf,
rem_bounds_absval,
set_wf,
equal_wf,
left_mul_subtract_distrib,
left_mul_add_distrib,
add_functionality_wrt_eq,
div_rem_sum2,
mul-distributes-right,
add-associates,
minus-add,
minus-minus,
mul-associates,
minus-one-mul,
mul-commutes,
zero-mul,
add-zero,
zero-add,
add-swap,
add-mul-special,
add-commutes,
le_functionality,
le_transitivity,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
absval_sym,
sq_stable__le,
sq_stable__less_than,
decidable__le,
intformnot_wf,
intformle_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
voidElimination,
voidEquality,
setEquality,
functionEquality,
intEquality,
functionExtensionality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
addEquality,
hyp_replacement,
dependent_pairFormation,
lambdaFormation,
divideEquality,
int_eqEquality,
computeAll,
multiplyEquality,
baseApply,
closedConclusion,
remainderEquality,
equalityUniverse,
levelHypothesis,
minusEquality,
unionElimination
Latex:
\mforall{}[x,y,z:\mBbbR{}]. ((x * (y + z)) = ((x * y) + (x * z)))
Date html generated:
2016_10_26-AM-09_03_57
Last ObjectModification:
2016_07_12-AM-08_14_35
Theory : reals
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