Theorem galactghm 19373

Description: The currying of a group action is a group homomorphism between the group 𝐺 and the symmetric group (SymGrp‘𝑌). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
galactghm.x	⊢ 𝑋 = (Base‘𝐺)
galactghm.h	⊢ 𝐻 = (SymGrp‘𝑌)
galactghm.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)))

Assertion

Ref	Expression
galactghm	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥, ⊕ ,𝑦   𝑥,𝑋,𝑦   𝑥,𝐻   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐻(𝑦)

Proof of Theorem galactghm

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		galactghm.x	. 2 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2741	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2741	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2741	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		gagrp 19261	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
6		gaset 19262	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
7		galactghm.h	. . . 4 ⊢ 𝐻 = (SymGrp‘𝑌)
8	7	symggrp 19369	. . 3 ⊢ (𝑌 ∈ V → 𝐻 ∈ Grp)
9	6, 8	syl 17	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐻 ∈ Grp)
10		eqid 2741	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦))
11	1, 10	gapm 19275	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌)
12	6	adantr 482	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
13	7, 2	elsymgbas 19343	. . . . 5 ⊢ (𝑌 ∈ V → ((𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻) ↔ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌))
14	12, 13	syl 17	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻) ↔ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌))
15	11, 14	mpbird 259	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻))
16		galactghm.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)))
17	15, 16	fmptd 7058	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹:𝑋⟶(Base‘𝐻))
18		df-3an 1095	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ↔ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌))
19	1, 3	gaass 19266	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
20	18, 19	sylan2br 602	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
21	20	anassrs 469	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
22	21	mpteq2dva 5167	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
23		oveq1 7366	. . . . 5 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑥 ⊕ 𝑦) = ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦))
24	23	mpteq2dv 5168	. . . 4 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
25	5	adantr 482	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐺 ∈ Grp)
26		simprl 777	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27		simprr 779	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
28	1, 3	grpcl 18912	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
29	25, 26, 27, 28	syl3anc 1380	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
30	6	adantr 482	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑌 ∈ V)
31	30	mptexd 7171	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) ∈ V)
32	16, 24, 29, 31	fvmptd3 6962	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝐺)𝑤)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
33	17	adantr 482	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
34	33, 26	ffvelcdmd 7029	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
35	33, 27	ffvelcdmd 7029	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ (Base‘𝐻))
36	7, 2, 4	symgov 19353	. . . . 5 ⊢ (((𝐹‘𝑧) ∈ (Base‘𝐻) ∧ (𝐹‘𝑤) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
37	34, 35, 36	syl2anc 591	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
38	1	gaf 19264	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
39	38	ad2antrr 733	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
40	27	adantr 482	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑤 ∈ 𝑋)
41		simpr 486	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
42	39, 40, 41	fovcdmd 7531	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → (𝑤 ⊕ 𝑦) ∈ 𝑌)
43		oveq1 7366	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 ⊕ 𝑦) = (𝑤 ⊕ 𝑦))
44	43	mpteq2dv 5168	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
45	30	mptexd 7171	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)) ∈ V)
46	16, 44, 27, 45	fvmptd3 6962	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
47		oveq1 7366	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ⊕ 𝑦) = (𝑧 ⊕ 𝑦))
48	47	mpteq2dv 5168	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
49	30	mptexd 7171	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) ∈ V)
50	16, 48, 26, 49	fvmptd3 6962	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
51		oveq2 7367	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ⊕ 𝑦) = (𝑧 ⊕ 𝑥))
52	51	cbvmptv 5178	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥))
53	50, 52	eqtrdi 2792	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥)))
54		oveq2 7367	. . . . 5 ⊢ (𝑥 = (𝑤 ⊕ 𝑦) → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
55	42, 46, 53, 54	fmptco 7074	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) ∘ (𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
56	37, 55	eqtrd 2776	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
57	22, 32, 56	3eqtr4d 2786	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝐺)𝑤)) = ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)))
58	1, 2, 3, 4, 5, 9, 17, 57	isghmd 19194	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ↦ cmpt 5155   × cxp 5618   ∘ ccom 5624  ⟶wf 6484  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  Grpcgrp 18904   GrpHom cghm 19182   GrpAct cga 19258  SymGrpcsymg 19338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-tset 17234  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-efmnd 18832  df-grp 18907  df-minusg 18908  df-ghm 19183  df-ga 19259  df-symg 19339

This theorem is referenced by:  cayleylem1  19381