Nuprl Lemma : rcp-distrib1
∀[a,b,c:ℝ^3]. req-vec(3;(a x b + c);(a x b) + (a x c))
Proof
Definitions occuring in Statement :
rcp: (a x b),
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
rcp: (a x b),
real-vec-add: X + Y,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
select: L[n],
cons: [a / b],
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
subtract: n - m,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
nat: ℕ,
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
false_wf,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
req_witness,
rcp_wf,
real-vec-add_wf,
le_wf,
real-vec_wf,
rsub_wf,
rmul_wf,
lelt_wf,
radd_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
addEquality,
independent_pairFormation,
productElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[a,b,c:\mBbbR{}\^{}3]. req-vec(3;(a x b + c);(a x b) + (a x c))
Date html generated:
2018_05_22-PM-02_41_17
Last ObjectModification:
2018_05_09-PM-01_11_33
Theory : reals
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