Theorem galactghm 19645

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 2763	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2763	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2763	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		gagrp 19533	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
6		gaset 19534	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
7		galactghm.h	. . . 4 ⊢ 𝐻 = (SymGrp‘𝑌)
8	7	symggrp 19641	. . 3 ⊢ (𝑌 ∈ V → 𝐻 ∈ Grp)
9	6, 8	syl 18	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐻 ∈ Grp)
10		eqid 2763	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦))
11	1, 10	gapm 19547	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌⟶1-1onto𝑌)
12	6	adantr 486	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
13	7, 2	elsymgbas 19615	. . . . 5 ⊢ (𝑌 ∈ V → ((𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻) ↔ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌⟶1-1onto𝑌))
14	12, 13	syl 18	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻) ↔ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌⟶1-1onto𝑌))
15	11, 14	mpbird 260	. . 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 7135	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹:𝑋⟶(Base‘𝐻))
18		df-3an 1105	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ↔ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌))
19	1, 3	gaass 19538	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
20	18, 19	sylan2br 607	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
21	20	anassrs 473	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
22	21	mpteq2dva 5201	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
23		oveq1 7447	. . . . 5 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑥 ⊕ 𝑦) = ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦))
24	23	mpteq2dv 5202	. . . 4 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
25	5	adantr 486	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐺 ∈ Grp)
26		simprl 783	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27		simprr 785	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
28	1, 3	grpcl 19178	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
29	25, 26, 27, 28	syl3anc 1399	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
30	6	adantr 486	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑌 ∈ V)
31	30	mptexd 7250	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) ∈ V)
32	16, 24, 29, 31	fvmptd3 7037	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝐺)𝑤)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
33	17	adantr 486	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
34	33, 26	ffvelcdmd 7106	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
35	33, 27	ffvelcdmd 7106	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ (Base‘𝐻))
36	7, 2, 4	symgov 19625	. . . . 5 ⊢ (((𝐹‘𝑧) ∈ (Base‘𝐻) ∧ (𝐹‘𝑤) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
37	34, 35, 36	syl2anc 596	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
38	1	gaf 19536	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
39	38	ad2antrr 739	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
40	27	adantr 486	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑤 ∈ 𝑋)
41		simpr 490	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
42	39, 40, 41	fovcdmd 7614	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → (𝑤 ⊕ 𝑦) ∈ 𝑌)
43		oveq1 7447	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 ⊕ 𝑦) = (𝑤 ⊕ 𝑦))
44	43	mpteq2dv 5202	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
45	30	mptexd 7250	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)) ∈ V)
46	16, 44, 27, 45	fvmptd3 7037	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
47		oveq1 7447	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ⊕ 𝑦) = (𝑧 ⊕ 𝑦))
48	47	mpteq2dv 5202	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
49	30	mptexd 7250	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) ∈ V)
50	16, 48, 26, 49	fvmptd3 7037	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
51		oveq2 7448	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ⊕ 𝑦) = (𝑧 ⊕ 𝑥))
52	51	cbvmptv 5212	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥))
53	50, 52	eqtrdi 2814	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥)))
54		oveq2 7448	. . . . 5 ⊢ (𝑥 = (𝑤 ⊕ 𝑦) → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
55	42, 46, 53, 54	fmptco 7151	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) ∘ (𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
56	37, 55	eqtrd 2798	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
57	22, 32, 56	3eqtr4d 2808	. 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 19465	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5189   × cxp 5653   ∘ ccom 5659  ⟶wf 6551  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  Grpcgrp 19170   GrpHom cghm 19453   GrpAct cga 19530  SymGrpcsymg 19610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-tset 17473  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-efmnd 19091  df-grp 19173  df-minusg 19174  df-ghm 19454  df-ga 19531  df-symg 19611

This theorem is used by:  cayleylem1  19653