Nuprl Lemma : module_over_triv_rng
∀A:Rng. ∀m:A-Module. ((1 = 0 ∈ |A|) ⇒ (∀u:m.car. (u = m.zero ∈ m.car)))
Proof
Definitions occuring in Statement :
module: A-Module,
alg_zero: a.zero,
alg_car: a.car,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
rng: Rng,
rng_one: 1,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
module: A-Module,
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
alg_car_wf,
rng_car_wf,
equal_wf,
rng_one_wf,
rng_zero_wf,
module_wf,
rng_wf,
infix_ap_wf,
alg_act_wf,
squash_wf,
true_wf,
module_action_p,
module_act_zero_l,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
because_Cache,
applyLambdaEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}A:Rng. \mforall{}m:A-Module. ((1 = 0) {}\mRightarrow{} (\mforall{}u:m.car. (u = m.zero)))
Date html generated:
2017_10_01-AM-09_51_53
Last ObjectModification:
2017_03_03-PM-00_46_32
Theory : algebras_1
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