Nuprl Lemma : q_distrib
∀[r,s,t:ℚ]. (((r + s) * t) = ((r * t) + (s * t)) ∈ ℚ)
Proof
Definitions occuring in Statement :
qmul: r * s,
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
qmul: r * s,
qadd: r + s,
qeq: qeq(r;s),
so_lambda: λ2x.t[x],
so_apply: x[s],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
assert-qeq,
qmul_wf,
qadd_wf,
q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
int-subtype-rationals,
equal-wf-base,
rationals_wf,
int_subtype_base,
istype-assert,
qdiv-int-elim,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nequal_wf,
valueall-type-has-valueall,
product-valueall-type,
int-valueall-type,
evalall-reduce,
assert_of_eq_int,
decidable__equal_int,
intformnot_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
qdiv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
lambdaFormation_alt,
independent_functionElimination,
applyEquality,
sqequalRule,
closedConclusion,
natural_numberEquality,
baseClosed,
because_Cache,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
equalityIstype,
inhabitedIsType,
sqequalBase,
equalitySymmetry,
intEquality,
productEquality,
independent_pairEquality,
callbyvalueReduce,
addEquality,
multiplyEquality,
unionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[r,s,t:\mBbbQ{}]. (((r + s) * t) = ((r * t) + (s * t)))
Date html generated:
2020_05_20-AM-09_13_34
Last ObjectModification:
2020_01_24-PM-03_55_37
Theory : rationals
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