Nuprl Lemma : quot_grp_inv_wf
∀g:IGroup. ∀h:NormSubGrp{i}(g). (~ ∈ |g//h| ⟶ |g//h|)
Proof
Definitions occuring in Statement :
quot_grp_car: |g//h|,
norm_subgrp: NormSubGrp{i}(g),
igrp: IGroup,
grp_inv: ~,
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
norm_subgrp: NormSubGrp{i}(g),
uall: ∀[x:A]. B[x],
igrp: IGroup,
imon: IMonoid,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
quot_grp_car: |g//h|,
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
guard: {T},
subgrp_p: s SubGrp of g,
eqv_mod_subset: a ≡ b (mod s in g),
norm_subset_p: norm_subset_p(g;s),
infix_ap: x f y,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
norm_subgrp_wf,
igrp_wf,
grp_inv_wf,
quot_grp_car_wf,
subgrp_p_wf,
grp_car_wf,
norm_subset_p_wf,
eqv_mod_subset_wf,
equal_wf,
equal-wf-base,
quotient-member-eq,
eqv_mod_subset_is_eqv,
grp_op_wf,
infix_ap_wf,
squash_wf,
true_wf,
grp_inv_thru_op,
grp_inv_inv,
mon_assoc,
grp_inv_assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
functionExtensionality,
productElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
applyEquality,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalRule,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaEquality,
independent_isectElimination,
hyp_replacement,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:IGroup. \mforall{}h:NormSubGrp\{i\}(g). (\msim{} \mmember{} |g//h| {}\mrightarrow{} |g//h|)
Date html generated:
2017_10_01-AM-08_13_55
Last ObjectModification:
2017_02_28-PM-01_58_29
Theory : groups_1
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