Nuprl Lemma : ringeq_int_terms_functionality
∀[r:Rng]. ∀[x1,x2,y1,y2:int_term()]. (uiff(x1 ≡ y1;x2 ≡ y2)) supposing (y1 ≡ y2 and x1 ≡ x2)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
rng: Rng,
int_term: int_term(),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
prop: ℙ,
rng: Rng,
all: ∀x:A. B[x],
ringeq_int_terms: t1 ≡ t2,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T
Lemmas referenced :
rng_wf,
int_term_wf,
ringeq_int_terms_wf,
rng_car_wf,
iff_weakening_equal,
ring_term_value_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
independent_pairEquality,
productElimination,
rename,
setElimination,
isectElimination,
extract_by_obid,
intEquality,
functionEquality,
axiomEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
because_Cache,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
functionExtensionality,
universeEquality,
imageElimination,
applyEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[x1,x2,y1,y2:int\_term()]. (uiff(x1 \mequiv{} y1;x2 \mequiv{} y2)) supposing (y1 \mequiv{} y2 and x1 \mequiv{} x2)
Date html generated:
2018_05_21-PM-03_15_53
Last ObjectModification:
2018_01_25-PM-01_30_08
Theory : rings_1
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