Nuprl Lemma : itermSubtract_functionality_wrt_ringeq
∀[r:Rng]. ∀[a,b,c,d:int_term()]. (a (-) c ≡ b (-) d) supposing (a ≡ b and c ≡ d)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
rng: Rng,
itermSubtract: left (-) right,
int_term: int_term(),
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
infix_ap: x f y,
rng: Rng,
prop: ℙ,
squash: ↓T,
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
ring_term_value: ring_term_value(f;t),
all: ∀x:A. B[x],
ringeq_int_terms: t1 ≡ t2,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
int_term_wf,
ringeq_int_terms_wf,
iff_weakening_equal,
infix_ap_wf,
rng_minus_wf,
rng_plus_wf,
rng_car_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
isect_memberEquality,
axiomEquality,
intEquality,
functionEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
rename,
setElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
isectElimination,
extract_by_obid,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b,c,d:int\_term()]. (a (-) c \mequiv{} b (-) d) supposing (a \mequiv{} b and c \mequiv{} d)
Date html generated:
2018_05_21-PM-03_16_10
Last ObjectModification:
2018_01_25-PM-02_18_59
Theory : rings_1
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