### Nuprl Lemma : fg-hom_wf

`∀[X:Type]. ∀[G:Group{i}]. ∀[f:X ⟶ |G|]. ∀[w:free-word(X)].  (fg-hom(G;f;w) ∈ |G|)`

Proof

Definitions occuring in Statement :  fg-hom: `fg-hom(G;f;w)` free-word: `free-word(X)` uall: `∀[x:A]. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` universe: `Type` grp: `Group{i}` grp_car: `|g|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` grp: `Group{i}` mon: `Mon` all: `∀x:A. B[x]` implies: `P `` Q` transitive-reflexive-closure: `R^*` or: `P ∨ Q` prop: `ℙ` fg-hom: `fg-hom(G;f;w)` squash: `↓T` so_lambda: `λ2x y.t[x; y]` infix_ap: `x f y` true: `True` so_apply: `x[s1;s2]` guard: `{T}` rel_implies: `R1 => R2` word-rel: `word-rel(X;w1;w2)` exists: `∃x:A. B[x]` and: `P ∧ Q` label: `...\$L... t` subtype_rel: `A ⊆r B` uimplies: `b supposing a` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` inverse-letters: `a = -b` top: `Top` imon: `IMonoid` trans: `Trans(T;x,y.E[x; y])` free-word: `free-word(X)` cand: `A c∧ B` word-equiv: `word-equiv(X;w1;w2)` quotient: `x,y:A//B[x; y]` equiv_rel: `EquivRel(T;x,y.E[x; y])` refl: `Refl(T;x,y.E[x; y])`
Lemmas referenced :  free-word_wf grp_car_wf grp_wf transitive-reflexive-closure_wf list_wf word-rel_wf list_accum_wf squash_wf true_wf grp_id_wf grp_op_wf grp_inv_wf equal_wf transitive-closure-minimal member_wf and_wf append_wf cons_wf nil_wf subtype_rel_self iff_weakening_equal fg-hom-append list_accum_cons_lemma list_accum_nil_lemma grp_sig_wf monoid_p_wf inverse_wf grp_subtype_igrp mon_assoc grp_inverse mon_ident word-equiv_wf word-equiv-equiv equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination thin hypothesisEquality isect_memberEquality because_Cache functionEquality setElimination rename universeEquality lambdaFormation unionElimination applyEquality unionEquality lambdaEquality imageElimination dependent_functionElimination independent_functionElimination natural_numberEquality imageMemberEquality baseClosed productElimination dependent_set_memberEquality independent_pairFormation hyp_replacement applyLambdaEquality instantiate independent_isectElimination voidElimination voidEquality setEquality cumulativity promote_hyp pointwiseFunctionality pertypeElimination productEquality

Latex:
\mforall{}[X:Type].  \mforall{}[G:Group\{i\}].  \mforall{}[f:X  {}\mrightarrow{}  |G|].  \mforall{}[w:free-word(X)].    (fg-hom(G;f;w)  \mmember{}  |G|)

Date html generated: 2020_05_20-AM-08_23_06
Last ObjectModification: 2018_08_21-PM-02_02_49

Theory : free!groups

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