Theorem lactghmga 17824

Description: The converse of galactghm 17823. The uncurrying of a homomorphism into (SymGrp‘𝑌) is a group action. Thus, group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   ⊕ (𝑥,𝑦)

Proof of Theorem lactghmga

Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmgrp1 17662	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2		ghmgrp2 17663	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3		grpn0 17454	. . . 4 ⊢ (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4		lactghmga.h	. . . . . 6 ⊢ 𝐻 = (SymGrp‘𝑌)
5		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
6	4, 5	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑌 ∈ V → 𝐻 = ∅)
7	6	necon1ai 2821	. . . 4 ⊢ (𝐻 ≠ ∅ → 𝑌 ∈ V)
8	2, 3, 7	3syl 18	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
9	1, 8	jca 554	. 2 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
10		lactghmga.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
11		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
12	10, 11	ghmf 17664	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
13	12	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ (Base‘𝐻))
14	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
15	4, 11	elsymgbas 17802	. . . . . . . . . 10 ⊢ (𝑌 ∈ V → ((𝐹‘𝑥) ∈ (Base‘𝐻) ↔ (𝐹‘𝑥):𝑌–1-1-onto→𝑌))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (Base‘𝐻) ↔ (𝐹‘𝑥):𝑌–1-1-onto→𝑌))
17	13, 16	mpbid 222	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥):𝑌–1-1-onto→𝑌)
18		f1of 6137	. . . . . . . 8 ⊢ ((𝐹‘𝑥):𝑌–1-1-onto→𝑌 → (𝐹‘𝑥):𝑌⟶𝑌)
19	17, 18	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥):𝑌⟶𝑌)
20	19	ffvelrnda 6359	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
21	20	ralrimiva 2966	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
22	21	ralrimiva 2966	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
23		lactghmga.f	. . . . 5 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)‘𝑦))
24	23	fmpt2 7237	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌 ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌)
25	22, 24	sylib 208	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
26		eqid 2622	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
27	10, 26	grpidcl 17450	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
28	1, 27	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g‘𝐺) ∈ 𝑋)
29		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝐹‘𝑥) = (𝐹‘(0g‘𝐺)))
30	29	fveq1d 6193	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑦))
31		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘(0g‘𝐺))‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑧))
32		fvex 6201	. . . . . . . 8 ⊢ ((𝐹‘(0g‘𝐺))‘𝑧) ∈ V
33	30, 31, 23, 32	ovmpt2 6796	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
34	28, 33	sylan 488	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
35		eqid 2622	. . . . . . . . . 10 ⊢ (0g‘𝐻) = (0g‘𝐻)
36	26, 35	ghmid 17666	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
38	8	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ V)
39	4	symgid 17821	. . . . . . . . 9 ⊢ (𝑌 ∈ V → ( I ↾ 𝑌) = (0g‘𝐻))
40	38, 39	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ( I ↾ 𝑌) = (0g‘𝐻))
41	37, 40	eqtr4d 2659	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = ( I ↾ 𝑌))
42	41	fveq1d 6193	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘(0g‘𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
43		fvresi 6439	. . . . . . 7 ⊢ (𝑧 ∈ 𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
44	43	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
45	34, 42, 44	3eqtrd 2660	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = 𝑧)
46	12	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
47		simprr 796	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑣 ∈ 𝑋)
48	46, 47	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣) ∈ (Base‘𝐻))
49	8	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑌 ∈ V)
50	4, 11	elsymgbas 17802	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ V → ((𝐹‘𝑣) ∈ (Base‘𝐻) ↔ (𝐹‘𝑣):𝑌–1-1-onto→𝑌))
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑣) ∈ (Base‘𝐻) ↔ (𝐹‘𝑣):𝑌–1-1-onto→𝑌))
52	48, 51	mpbid 222	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣):𝑌–1-1-onto→𝑌)
53		f1of 6137	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣):𝑌–1-1-onto→𝑌 → (𝐹‘𝑣):𝑌⟶𝑌)
54	52, 53	syl 17	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣):𝑌⟶𝑌)
55		simplr 792	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑧 ∈ 𝑌)
56		fvco3 6275	. . . . . . . . 9 ⊢ (((𝐹‘𝑣):𝑌⟶𝑌 ∧ 𝑧 ∈ 𝑌) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
57	54, 55, 56	syl2anc 693	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
58		simpll 790	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
59		simprl 794	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑢 ∈ 𝑋)
60		eqid 2622	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
61		eqid 2622	. . . . . . . . . . . 12 ⊢ (+g‘𝐻) = (+g‘𝐻)
62	10, 60, 61	ghmlin 17665	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)))
63	58, 59, 47, 62	syl3anc 1326	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)))
64	46, 59	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑢) ∈ (Base‘𝐻))
65	4, 11, 61	symgov 17810	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑢) ∈ (Base‘𝐻) ∧ (𝐹‘𝑣) ∈ (Base‘𝐻)) → ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
66	64, 48, 65	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
67	63, 66	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
68	67	fveq1d 6193	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) = (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧))
69	54, 55	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑣)‘𝑧) ∈ 𝑌)
70		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
71	70	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑢)‘𝑦))
72		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = ((𝐹‘𝑣)‘𝑧) → ((𝐹‘𝑢)‘𝑦) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
73		fvex 6201	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)) ∈ V
74	71, 72, 23, 73	ovmpt2 6796	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ((𝐹‘𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
75	59, 69, 74	syl2anc 693	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
76	57, 68, 75	3eqtr4d 2666	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
77	1	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐺 ∈ Grp)
78	10, 60	grpcl 17430	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
79	77, 59, 47, 78	syl3anc 1326	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
80		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → (𝐹‘𝑥) = (𝐹‘(𝑢(+g‘𝐺)𝑣)))
81	80	fveq1d 6193	. . . . . . . . 9 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦))
82		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
83		fvex 6201	. . . . . . . . 9 ⊢ ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) ∈ V
84	81, 82, 23, 83	ovmpt2 6796	. . . . . . . 8 ⊢ (((𝑢(+g‘𝐺)𝑣) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
85	79, 55, 84	syl2anc 693	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
86		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
87	86	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑣)‘𝑦))
88		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑣)‘𝑦) = ((𝐹‘𝑣)‘𝑧))
89		fvex 6201	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣)‘𝑧) ∈ V
90	87, 88, 23, 89	ovmpt2 6796	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
91	47, 55, 90	syl2anc 693	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
92	91	oveq2d 6666	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 ⊕ (𝑣 ⊕ 𝑧)) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
93	76, 85, 92	3eqtr4d 2666	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
94	93	ralrimivva 2971	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
95	45, 94	jca 554	. . . 4 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
96	95	ralrimiva 2966	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
97	25, 96	jca 554	. 2 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))))
98	10, 60, 26	isga 17724	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))))
99	9, 97, 98	sylanbrc 698	1 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ ∈ (𝐺 GrpAct 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ∅c0 3915   I cid 5023   × cxp 5112   ↾ cres 5116   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422   GrpHom cghm 17657   GrpAct cga 17722  SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-ga 17723  df-symg 17798

This theorem is referenced by:  symgga  17826