Theorem lactghmga 19346

Description: The converse of galactghm 19345. 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 19159	. 2 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2		ghmgrp2 19160	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3		grpn0 18913	. . 3 ⊢ (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4		lactghmga.h	. . . . 5 ⊢ 𝐻 = (SymGrp‘𝑌)
5		fvprc 6834	. . . . 5 ⊢ (¬ 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
6	4, 5	eqtrid 2784	. . . 4 ⊢ (¬ 𝑌 ∈ V → 𝐻 = ∅)
7	6	necon1ai 2960	. . 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 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
11	9, 10	ghmf 19161	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
12	11	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ (Base‘𝐻))
13	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ V)
14	4, 10	elsymgbas 19315	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥):𝑌–1-1-onto→𝑌)
17		f1of 6782	. . . . . . . 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	ffvelcdmda 7038	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
20	19	ralrimiva 3130	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌)
21	20	ralrimiva 3130	. . . 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 8022	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)‘𝑦) ∈ 𝑌 ↔ ⊕ :(𝑋 × 𝑌)⟶𝑌)
24	21, 23	sylib 218	. . 3 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
25		eqid 2737	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
26	9, 25	grpidcl 18907	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
27	1, 26	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g‘𝐺) ∈ 𝑋)
28		fveq2 6842	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝐹‘𝑥) = (𝐹‘(0g‘𝐺)))
29	28	fveq1d 6844	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑦))
30		fveq2 6842	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘(0g‘𝐺))‘𝑦) = ((𝐹‘(0g‘𝐺))‘𝑧))
31		fvex 6855	. . . . . . . 8 ⊢ ((𝐹‘(0g‘𝐺))‘𝑧) ∈ V
32	29, 30, 22, 31	ovmpo 7528	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
33	27, 32	sylan 581	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = ((𝐹‘(0g‘𝐺))‘𝑧))
34		eqid 2737	. . . . . . . . . 10 ⊢ (0g‘𝐻) = (0g‘𝐻)
35	25, 34	ghmid 19163	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻))
37	8	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ V)
38	4	symgid 19342	. . . . . . . . 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 2775	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (𝐹‘(0g‘𝐺)) = ( I ↾ 𝑌))
41	40	fveq1d 6844	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘(0g‘𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
42		fvresi 7129	. . . . . . 7 ⊢ (𝑧 ∈ 𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
43	42	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
44	33, 41, 43	3eqtrd 2776	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = 𝑧)
45	11	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
46		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑣 ∈ 𝑋)
47	45, 46	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣) ∈ (Base‘𝐻))
48	8	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑌 ∈ V)
49	4, 10	elsymgbas 19315	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑣):𝑌–1-1-onto→𝑌)
52		f1of 6782	. . . . . . . . . 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 769	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑧 ∈ 𝑌)
55		fvco3 6941	. . . . . . . . 9 ⊢ (((𝐹‘𝑣):𝑌⟶𝑌 ∧ 𝑧 ∈ 𝑌) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
56	53, 54, 55	syl2anc 585	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (((𝐹‘𝑢) ∘ (𝐹‘𝑣))‘𝑧) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
57		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
58		simprl 771	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑢 ∈ 𝑋)
59		eqid 2737	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
60		eqid 2737	. . . . . . . . . . . 12 ⊢ (+g‘𝐻) = (+g‘𝐻)
61	9, 59, 60	ghmlin 19162	. . . . . . . . . . 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 1374	. . . . . . . . . 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	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘𝑢) ∈ (Base‘𝐻))
64	4, 10, 60	symgov 19325	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑢) ∈ (Base‘𝐻) ∧ (𝐹‘𝑣) ∈ (Base‘𝐻)) → ((𝐹‘𝑢)(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
65	63, 47, 64	syl2anc 585	. . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝐹‘(𝑢(+g‘𝐺)𝑣)) = ((𝐹‘𝑢) ∘ (𝐹‘𝑣)))
67	66	fveq1d 6844	. . . . . . . 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	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘𝑣)‘𝑧) ∈ 𝑌)
69		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
70	69	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑢)‘𝑦))
71		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = ((𝐹‘𝑣)‘𝑧) → ((𝐹‘𝑢)‘𝑦) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
72		fvex 6855	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)) ∈ V
73	70, 71, 22, 72	ovmpo 7528	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ((𝐹‘𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)) = ((𝐹‘𝑢)‘((𝐹‘𝑣)‘𝑧)))
74	58, 68, 73	syl2anc 585	. . . . . . . 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 2782	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
76	1	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐺 ∈ Grp)
77	9, 59	grpcl 18883	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
78	76, 58, 46, 77	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
79		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → (𝐹‘𝑥) = (𝐹‘(𝑢(+g‘𝐺)𝑣)))
80	79	fveq1d 6844	. . . . . . . . 9 ⊢ (𝑥 = (𝑢(+g‘𝐺)𝑣) → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦))
81		fveq2 6842	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
82		fvex 6855	. . . . . . . . 9 ⊢ ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧) ∈ V
83	80, 81, 22, 82	ovmpo 7528	. . . . . . . 8 ⊢ (((𝑢(+g‘𝐺)𝑣) ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
84	78, 54, 83	syl2anc 585	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = ((𝐹‘(𝑢(+g‘𝐺)𝑣))‘𝑧))
85		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
86	85	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → ((𝐹‘𝑥)‘𝑦) = ((𝐹‘𝑣)‘𝑦))
87		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑣)‘𝑦) = ((𝐹‘𝑣)‘𝑧))
88		fvex 6855	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣)‘𝑧) ∈ V
89	86, 87, 22, 88	ovmpo 7528	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
90	46, 54, 89	syl2anc 585	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ⊕ 𝑧) = ((𝐹‘𝑣)‘𝑧))
91	90	oveq2d 7384	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢 ⊕ (𝑣 ⊕ 𝑧)) = (𝑢 ⊕ ((𝐹‘𝑣)‘𝑧)))
92	75, 84, 91	3eqtr4d 2782	. . . . . 6 ⊢ (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
93	92	ralrimivva 3181	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ 𝑌) → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
94	44, 93	jca 511	. . . 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 3130	. . 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 511	. 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 19232	. 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 1346	1 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ⊕ ∈ (𝐺 GrpAct 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442  ∅c0 4287   I cid 5526   × cxp 5630   ↾ cres 5634   ∘ ccom 5636  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Grpcgrp 18875   GrpHom cghm 19153   GrpAct cga 19230  SymGrpcsymg 19310

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-tset 17208  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-efmnd 18806  df-grp 18878  df-ghm 19154  df-ga 19231  df-symg 19311

This theorem is referenced by:  symgga  19348