Theorem galactghm 19473

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 2761	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2761	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2761	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		gagrp 19361	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
6		gaset 19362	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
7		galactghm.h	. . . 4 ⊢ 𝐻 = (SymGrp‘𝑌)
8	7	symggrp 19469	. . 3 ⊢ (𝑌 ∈ V → 𝐻 ∈ Grp)
9	6, 8	syl 18	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐻 ∈ Grp)
10		eqid 2761	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦))
11	1, 10	gapm 19375	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌)
12	6	adantr 485	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
13	7, 2	elsymgbas 19443	. . . . 5 ⊢ (𝑌 ∈ V → ((𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) ∈ (Base‘𝐻) ↔ (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)):𝑌–1-1-onto→𝑌))
14	12, 13	syl 18	. . . 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 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 7109	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹:𝑋⟶(Base‘𝐻))
18		df-3an 1103	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ↔ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌))
19	1, 3	gaass 19366	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
20	18, 19	sylan2br 606	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌)) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
21	20	anassrs 472	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
22	21	mpteq2dva 5203	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
23		oveq1 7417	. . . . 5 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑥 ⊕ 𝑦) = ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦))
24	23	mpteq2dv 5204	. . . 4 ⊢ (𝑥 = (𝑧(+g‘𝐺)𝑤) → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
25	5	adantr 485	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐺 ∈ Grp)
26		simprl 782	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27		simprr 784	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑤 ∈ 𝑋)
28	1, 3	grpcl 19007	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
29	25, 26, 27, 28	syl3anc 1396	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧(+g‘𝐺)𝑤) ∈ 𝑋)
30	6	adantr 485	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑌 ∈ V)
31	30	mptexd 7222	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)) ∈ V)
32	16, 24, 29, 31	fvmptd3 7013	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝐺)𝑤)) = (𝑦 ∈ 𝑌 ↦ ((𝑧(+g‘𝐺)𝑤) ⊕ 𝑦)))
33	17	adantr 485	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
34	33, 26	ffvelcdmd 7080	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
35	33, 27	ffvelcdmd 7080	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) ∈ (Base‘𝐻))
36	7, 2, 4	symgov 19453	. . . . 5 ⊢ (((𝐹‘𝑧) ∈ (Base‘𝐻) ∧ (𝐹‘𝑤) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
37	34, 35, 36	syl2anc 595	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = ((𝐹‘𝑧) ∘ (𝐹‘𝑤)))
38	1	gaf 19364	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
39	38	ad2antrr 738	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
40	27	adantr 485	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑤 ∈ 𝑋)
41		simpr 489	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
42	39, 40, 41	fovcdmd 7582	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑌) → (𝑤 ⊕ 𝑦) ∈ 𝑌)
43		oveq1 7417	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 ⊕ 𝑦) = (𝑤 ⊕ 𝑦))
44	43	mpteq2dv 5204	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
45	30	mptexd 7222	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)) ∈ V)
46	16, 44, 27, 45	fvmptd3 7013	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑤) = (𝑦 ∈ 𝑌 ↦ (𝑤 ⊕ 𝑦)))
47		oveq1 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ⊕ 𝑦) = (𝑧 ⊕ 𝑦))
48	47	mpteq2dv 5204	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑌 ↦ (𝑥 ⊕ 𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
49	30	mptexd 7222	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) ∈ V)
50	16, 48, 26, 49	fvmptd3 7013	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)))
51		oveq2 7418	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧 ⊕ 𝑦) = (𝑧 ⊕ 𝑥))
52	51	cbvmptv 5214	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑦)) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥))
53	50, 52	eqtrdi 2812	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘𝑧) = (𝑥 ∈ 𝑌 ↦ (𝑧 ⊕ 𝑥)))
54		oveq2 7418	. . . . 5 ⊢ (𝑥 = (𝑤 ⊕ 𝑦) → (𝑧 ⊕ 𝑥) = (𝑧 ⊕ (𝑤 ⊕ 𝑦)))
55	42, 46, 53, 54	fmptco 7125	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) ∘ (𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
56	37, 55	eqtrd 2796	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝐻)(𝐹‘𝑤)) = (𝑦 ∈ 𝑌 ↦ (𝑧 ⊕ (𝑤 ⊕ 𝑦))))
57	22, 32, 56	3eqtr4d 2806	. 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 19294	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ↦ cmpt 5191   × cxp 5659   ∘ ccom 5665  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  Grpcgrp 18999   GrpHom cghm 19282   GrpAct cga 19358  SymGrpcsymg 19438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-tset 17328  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-efmnd 18927  df-grp 19002  df-minusg 19003  df-ghm 19283  df-ga 19359  df-symg 19439

This theorem is referenced by:  cayleylem1  19481