cayleylem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cayleylem1		Structured version Visualization version GIF version

Theorem cayleylem1 18535

Description: Lemma for cayley 18537. (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 2820	. . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))
4	1, 2, 3	gaid2 18428	. 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 7158	. . . . . . 7 ⊢ ((𝑥 = 𝑔 ∧ 𝑦 = 𝑎) → (𝑥 + 𝑦) = (𝑔 + 𝑎))
8		ovex 7182	. . . . . . 7 ⊢ (𝑔 + 𝑎) ∈ V
9	7, 3, 8	ovmpoa 7298	. . . . . 6 ⊢ ((𝑔 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎) = (𝑔 + 𝑎))
10	9	mpteq2dva 5154	. . . . 5 ⊢ (𝑔 ∈ 𝑋 → (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
11	10	mpteq2ia 5150	. . . 4 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎))) = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎)))
12	6, 11	eqtr4i 2846	. . 3 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))𝑎)))
13	1, 5, 12	galactghm 18527	. 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 1536 ∈ wcel 2113 ↦ cmpt 5139 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 Basecbs 16478 +_gcplusg 16560 0_gc0g 16708 Grpcgrp 18098 GrpHom cghm 18350 GrpAct cga 18414 SymGrpcsymg 18490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-tset 16579 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-efmnd 18029 df-grp 18101 df-minusg 18102 df-subg 18271 df-ghm 18351 df-ga 18415 df-symg 18491

This theorem is referenced by: cayleylem2 18536 cayley 18537

W3C validator