Nuprl Lemma : test-ring-eq
∀[r:CRng]
∀v,v1,v2,v3,v4,v5,v6,v7,v8,v9:|r|.
((((((v6 * v3) +r ((v2 * v5) +r (v4 * v7))) +r (-r ((v7 * v2) +r ((v3 * v4) +r (v5 * v6)))))
*
(((v6 * v9) +r ((v8 * v1) +r (v * v7))) +r (-r ((v7 * v8) +r ((v9 * v) +r (v1 * v6))))))
+r
(((((v * v7) +r ((v6 * v5) +r (v4 * v1))) +r (-r ((v1 * v6) +r ((v7 * v4) +r (v5 * v)))))
*
(((v6 * v9) +r ((v8 * v3) +r (v2 * v7))) +r (-r ((v7 * v8) +r ((v9 * v2) +r (v3 * v6))))))
+r
((((v * v3) +r ((v2 * v7) +r (v6 * v1))) +r (-r ((v1 * v2) +r ((v3 * v6) +r (v7 * v)))))
*
(((v6 * v9) +r ((v8 * v5) +r (v4 * v7))) +r (-r ((v7 * v8) +r ((v9 * v4) +r (v5 * v6))))))))
= 0
∈ |r|)
Proof
Definitions occuring in Statement :
crng: CRng,
rng_times: *,
rng_minus: -r,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
infix_ap: x f y,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ringeq_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top
Lemmas referenced :
rng_car_wf,
crng_wf,
rng_plus_wf,
rng_times_wf,
rng_minus_wf,
rng_zero_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
istype-int,
ring_term_value_add_lemma,
istype-void,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_minus_lemma,
ring_term_value_const_lemma,
int-to-ring-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
hypothesis,
inhabitedIsType,
hypothesisEquality,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
applyEquality,
because_Cache,
natural_numberEquality,
productElimination,
independent_isectElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[r:CRng]
\mforall{}v,v1,v2,v3,v4,v5,v6,v7,v8,v9:|r|.
((((((v6 * v3) +r ((v2 * v5) +r (v4 * v7))) +r (-r ((v7 * v2) +r ((v3 * v4) +r (v5 * v6)))))
*
(((v6 * v9) +r ((v8 * v1) +r (v * v7))) +r (-r ((v7 * v8) +r ((v9 * v) +r (v1 * v6))))))
+r
(((((v * v7) +r ((v6 * v5) +r (v4 * v1))) +r (-r ((v1 * v6) +r ((v7 * v4) +r (v5 * v)))))
*
(((v6 * v9) +r ((v8 * v3) +r (v2 * v7))) +r (-r ((v7 * v8) +r ((v9 * v2) +r (v3 * v6))))))
+r
((((v * v3) +r ((v2 * v7) +r (v6 * v1))) +r (-r ((v1 * v2) +r ((v3 * v6) +r (v7 * v)))))
*
(((v6 * v9) +r ((v8 * v5) +r (v4 * v7))) +r (-r ((v7 * v8) +r ((v9 * v4) +r (v5 * v6))))))))
= 0)
Date html generated:
2019_10_15-AM-10_34_14
Last ObjectModification:
2018_10_08-PM-06_00_25
Theory : rings_1
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