Theorem galactghm 19376

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 2737	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2737	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2737	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		gagrp 19264	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
6		gaset 19265	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
7		galactghm.h	. . . 4 ⊢ 𝐻 = (SymGrp‘𝑌)
8	7	symggrp 19372	. . 3 ⊢ (𝑌 ∈ V → 𝐻 ∈ Grp)
9	6, 8	syl 17	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐻 ∈ Grp)
10		eqid 2737	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦))
11	1, 10	gapm 19278	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌)
12	6	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
13	7, 2	elsymgbas 19346	. . . . 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 257	. . 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 7064	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹:𝑋⟶(Base‘𝐻))
18		df-3an 1089	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ↔ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌))
19	1, 3	gaass 19269	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
20	18, 19	sylan2br 596	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
21	20	anassrs 467	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
22	21	mpteq2dva 5179	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
23		oveq1 7371	. . . . 5 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑥 ⊕ 𝑦) = ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦))
24	23	mpteq2dv 5180	. . . 4 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
25	5	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐺 ∈ Grp)
26		simprl 771	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27		simprr 773	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
28	1, 3	grpcl 18914	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
29	25, 26, 27, 28	syl3anc 1374	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
30	6	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑌 ∈ V)
31	30	mptexd 7176	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) ∈ V)
32	16, 24, 29, 31	fvmptd3 6969	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝐺)𝑤)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
33	17	adantr 480	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
34	33, 26	ffvelcdmd 7035	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
35	33, 27	ffvelcdmd 7035	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ (Base‘𝐻))
36	7, 2, 4	symgov 19356	. . . . 5 ⊢ (((𝐹‘𝑧) ∈ (Base‘𝐻) ∧ (𝐹‘𝑤) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
37	34, 35, 36	syl2anc 585	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
38	1	gaf 19267	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
39	38	ad2antrr 727	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
40	27	adantr 480	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑤 ∈ 𝑋)
41		simpr 484	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
42	39, 40, 41	fovcdmd 7536	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → (𝑤 ⊕ 𝑦) ∈ 𝑌)
43		oveq1 7371	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 ⊕ 𝑦) = (𝑤 ⊕ 𝑦))
44	43	mpteq2dv 5180	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
45	30	mptexd 7176	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)) ∈ V)
46	16, 44, 27, 45	fvmptd3 6969	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
47		oveq1 7371	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ⊕ 𝑦) = (𝑧 ⊕ 𝑦))
48	47	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
49	30	mptexd 7176	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) ∈ V)
50	16, 48, 26, 49	fvmptd3 6969	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
51		oveq2 7372	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ⊕ 𝑦) = (𝑧 ⊕ 𝑥))
52	51	cbvmptv 5190	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥))
53	50, 52	eqtrdi 2788	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥)))
54		oveq2 7372	. . . . 5 ⊢ (𝑥 = (𝑤 ⊕ 𝑦) → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
55	42, 46, 53, 54	fmptco 7080	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) ∘ (𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
56	37, 55	eqtrd 2772	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
57	22, 32, 56	3eqtr4d 2782	. 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 19197	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ↦ cmpt 5167   × cxp 5626   ∘ ccom 5632  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7364  Basecbs 17176  +_gcplusg 17217  Grpcgrp 18906   GrpHom cghm 19184   GrpAct cga 19261  SymGrpcsymg 19341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-2 12241  df-3 12242  df-4 12243  df-5 12244  df-6 12245  df-7 12246  df-8 12247  df-9 12248  df-n0 12435  df-z 12522  df-uz 12786  df-fz 13459  df-struct 17114  df-sets 17131  df-slot 17149  df-ndx 17161  df-base 17177  df-ress 17198  df-plusg 17230  df-tset 17236  df-0g 17401  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-efmnd 18834  df-grp 18909  df-minusg 18910  df-ghm 19185  df-ga 19262  df-symg 19342

This theorem is referenced by:  cayleylem1  19384