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Theorem cayleylem1 19324

Description: Lemma for cayley 19326. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

**Hypotheses**
Ref	Expression
cayleylem1.x	⊢ 𝑋 = (Base‘𝐺)
cayleylem1.p	⊢ + = (+_g‘𝐺)
cayleylem1.u	⊢ 0 = (0_g‘𝐺)
cayleylem1.h	⊢ 𝐻 = (SymGrp‘𝑋)
cayleylem1.s	⊢ 𝑆 = (Base‘𝐻)
cayleylem1.f	⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))

**Assertion**
Ref	Expression
cayleylem1	⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Distinct variable groups: 𝑔,𝑎, + 𝐺,𝑎,𝑔 𝑔,𝐻 𝑋,𝑎,𝑔 0 ,𝑎 Allowed substitution hints: 𝑆(𝑔,𝑎) 𝐹(𝑔,𝑎) 𝐻(𝑎) 0 (𝑔)

Proof of Theorem cayleylem1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cayleylem1.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		cayleylem1.p	. . 3 ⊢ + = (+_g‘𝐺)
3		eqid 2731	. . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))
4	1, 2, 3	gaid2 19215	. 2 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ∈ (𝐺 GrpAct 𝑋))
5		cayleylem1.h	. . 3 ⊢ 𝐻 = (SymGrp‘𝑋)
6		cayleylem1.f	. . . 4 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
7		oveq12 7355	. . . . . . 7 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑦) = (𝑔 + 𝑎))
8		ovex 7379	. . . . . . 7 ⊢ (𝑔 + 𝑎) ∈ V
9	7, 3, 8	ovmpoa 7501	. . . . . 6 ⊢ ((𝑔 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎) = (𝑔 + 𝑎))
10	9	mpteq2dva 5182	. . . . 5 ⊢ (𝑔 ∈ 𝑋 → (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
11	10	mpteq2ia 5184	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
12	6, 11	eqtr4i 2757	. . 3 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎)))
13	1, 5, 12	galactghm 19316	. 2 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) ∈ (𝐺 GrpAct 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
14	4, 13	syl 17	1 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Grpcgrp 18846 GrpHom cghm 19124 GrpAct cga 19201 SymGrpcsymg 19281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-tset 17180 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-efmnd 18777 df-grp 18849 df-minusg 18850 df-subg 19036 df-ghm 19125 df-ga 19202 df-symg 19282

This theorem is referenced by: cayleylem2 19325 cayley 19326

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