Theorem lactghmga 18527

Description: The converse of galactghm 18526. 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 18354	. 2 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2		ghmgrp2 18355	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3		grpn0 18129	. . 3 ⊢ (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4		lactghmga.h	. . . . 5 ⊢ 𝐻 = (SymGrp‘𝑌)
5		fvprc 6657	. . . . 5 ⊢ (¬ 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
6	4, 5	syl5eq 2868	. . . 4 ⊢ (¬ 𝑌 ∈ V → 𝐻 = ∅)
7	6	necon1ai 3043	. . 3 ⊢ (𝐻 ≠ ∅ → 𝑌 ∈ V)
8	2, 3, 7	3syl 18	. 2 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
9		lactghmga.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
10		eqid 2821	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
11	9, 10	ghmf 18356	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
12	11	ffvelrnda 6845	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ (Base‘𝐻))
13	8	adantr 483	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
14	4, 10	elsymgbas 18496	. . . . . . . . . 10 ⊢ (𝑌 ∈ V → ((𝐹‘𝑥) ∈ (Base‘𝐻) ↔ (𝐹‘𝑥):𝑌–1-1-onto→𝑌))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (Base‘𝐻) ↔ (𝐹‘𝑥):𝑌–1-1-onto→𝑌))
16	12, 15	mpbid 234	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥):𝑌–1-1-onto→𝑌)
17		f1of 6609	. . . . . . . 8 ⊢ ((𝐹‘𝑥):𝑌–1-1-onto→𝑌 → (𝐹‘𝑥):𝑌⟶𝑌)
18	16, 17	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥):𝑌⟶𝑌)
19	18	ffvelrnda 6845	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
20	19	ralrimiva 3182	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
21	20	ralrimiva 3182	. . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
22		lactghmga.f	. . . . 5 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)‘𝑦))
23	22	fmpo 7760	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌 ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌)
24	21, 23	sylib 220	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
25		eqid 2821	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
26	9, 25	grpidcl 18125	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
27	1, 26	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g‘𝐺) ∈ 𝑋)
28		fveq2 6664	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝐹‘𝑥) = (𝐹‘(0g‘𝐺)))
29	28	fveq1d 6666	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑦))
30		fveq2 6664	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘(0g‘𝐺))‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑧))
31		fvex 6677	. . . . . . . 8 ⊢ ((𝐹‘(0g‘𝐺))‘𝑧) ∈ V
32	29, 30, 22, 31	ovmpo 7304	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
33	27, 32	sylan 582	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
34		eqid 2821	. . . . . . . . . 10 ⊢ (0g‘𝐻) = (0g‘𝐻)
35	25, 34	ghmid 18358	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
36	35	adantr 483	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
37	8	adantr 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ V)
38	4	symgid 18523	. . . . . . . . 9 ⊢ (𝑌 ∈ V → ( I ↾ 𝑌) = (0g‘𝐻))
39	37, 38	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ( I ↾ 𝑌) = (0g‘𝐻))
40	36, 39	eqtr4d 2859	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = ( I ↾ 𝑌))
41	40	fveq1d 6666	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘(0g‘𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
42		fvresi 6929	. . . . . . 7 ⊢ (𝑧 ∈ 𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
43	42	adantl 484	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
44	33, 41, 43	3eqtrd 2860	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = 𝑧)
45	11	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
46		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑣 ∈ 𝑋)
47	45, 46	ffvelrnd 6846	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣) ∈ (Base‘𝐻))
48	8	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑌 ∈ V)
49	4, 10	elsymgbas 18496	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ V → ((𝐹‘𝑣) ∈ (Base‘𝐻) ↔ (𝐹‘𝑣):𝑌–1-1-onto→𝑌))
50	48, 49	syl 17	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑣) ∈ (Base‘𝐻) ↔ (𝐹‘𝑣):𝑌–1-1-onto→𝑌))
51	47, 50	mpbid 234	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣):𝑌–1-1-onto→𝑌)
52		f1of 6609	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣):𝑌–1-1-onto→𝑌 → (𝐹‘𝑣):𝑌⟶𝑌)
53	51, 52	syl 17	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣):𝑌⟶𝑌)
54		simplr 767	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑧 ∈ 𝑌)
55		fvco3 6754	. . . . . . . . 9 ⊢ (((𝐹‘𝑣):𝑌⟶𝑌 ∧ 𝑧 ∈ 𝑌) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
56	53, 54, 55	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
57		simpll 765	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
58		simprl 769	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑢 ∈ 𝑋)
59		eqid 2821	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
60		eqid 2821	. . . . . . . . . . . 12 ⊢ (+g‘𝐻) = (+g‘𝐻)
61	9, 59, 60	ghmlin 18357	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)))
62	57, 58, 46, 61	syl3anc 1367	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)))
63	45, 58	ffvelrnd 6846	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑢) ∈ (Base‘𝐻))
64	4, 10, 60	symgov 18506	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑢) ∈ (Base‘𝐻) ∧ (𝐹‘𝑣) ∈ (Base‘𝐻)) → ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
65	63, 47, 64	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
66	62, 65	eqtrd 2856	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
67	66	fveq1d 6666	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) = (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧))
68	53, 54	ffvelrnd 6846	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑣)‘𝑧) ∈ 𝑌)
69		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
70	69	fveq1d 6666	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑢)‘𝑦))
71		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑦 = ((𝐹‘𝑣)‘𝑧) → ((𝐹‘𝑢)‘𝑦) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
72		fvex 6677	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)) ∈ V
73	70, 71, 22, 72	ovmpo 7304	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ((𝐹‘𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
74	58, 68, 73	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
75	56, 67, 74	3eqtr4d 2866	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
76	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐺 ∈ Grp)
77	9, 59	grpcl 18105	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
78	76, 58, 46, 77	syl3anc 1367	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
79		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → (𝐹‘𝑥) = (𝐹‘(𝑢(+g‘𝐺)𝑣)))
80	79	fveq1d 6666	. . . . . . . . 9 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦))
81		fveq2 6664	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
82		fvex 6677	. . . . . . . . 9 ⊢ ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) ∈ V
83	80, 81, 22, 82	ovmpo 7304	. . . . . . . 8 ⊢ (((𝑢(+g‘𝐺)𝑣) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
84	78, 54, 83	syl2anc 586	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
85		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
86	85	fveq1d 6666	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑣)‘𝑦))
87		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑣)‘𝑦) = ((𝐹‘𝑣)‘𝑧))
88		fvex 6677	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣)‘𝑧) ∈ V
89	86, 87, 22, 88	ovmpo 7304	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
90	46, 54, 89	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
91	90	oveq2d 7166	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 ⊕ (𝑣 ⊕ 𝑧)) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
92	75, 84, 91	3eqtr4d 2866	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
93	92	ralrimivva 3191	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
94	44, 93	jca 514	. . . 4 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
95	94	ralrimiva 3182	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
96	24, 95	jca 514	. 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)))))
97	9, 59, 25	isga 18415	. 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))))))
98	1, 8, 96, 97	syl21anbrc 1340	1 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ ∈ (𝐺 GrpAct 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  Vcvv 3494  ∅c0 4290   I cid 5453   × cxp 5547   ↾ cres 5551   ∘ ccom 5553  ⟶wf 6345  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Basecbs 16477  +_gcplusg 16559  0_gc0g 16707  Grpcgrp 18097   GrpHom cghm 18349   GrpAct cga 18413  SymGrpcsymg 18489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-tset 16578  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-efmnd 18028  df-grp 18100  df-ghm 18350  df-ga 18414  df-symg 18490

This theorem is referenced by:  symgga  18529