Theorem conjga 33774

Description: Group conjugation induces a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
cntrval2.1	⊢ 𝐵 = (Base‘𝑀)
cntrval2.2	⊢ + = (+g‘𝑀)
cntrval2.3	⊢ − = (-g‘𝑀)
cntrval2.4	⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))

Assertion

Ref	Expression
conjga	⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))

Distinct variable groups:   𝑥, ⊕ ,𝑦   𝑥, + ,𝑦   𝑥, − ,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦

Proof of Theorem conjga

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

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Grp)
2		cntrval2.1	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3	2	fvexi 6899	. . 3 ⊢ 𝐵 ∈ V
4	3	a1i 11	. 2 ⊢ (𝑀 ∈ Grp → 𝐵 ∈ V)
5		cntrval2.3	. . . 4 ⊢ − = (-g‘𝑀)
6	1	adantr 486	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ (𝐵 × 𝐵)) → 𝑀 ∈ Grp)
7		cntrval2.2	. . . . 5 ⊢ + = (+g‘𝑀)
8		xp1st 8033	. . . . . 6 ⊢ (𝑧 ∈ (𝐵 × 𝐵) → (1st ‘𝑧) ∈ 𝐵)
9	8	adantl 487	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ (𝐵 × 𝐵)) → (1st ‘𝑧) ∈ 𝐵)
10		xp2nd 8034	. . . . . 6 ⊢ (𝑧 ∈ (𝐵 × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
11	10	adantl 487	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ (𝐵 × 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
12	2, 7, 6, 9, 11	grpcld 19159	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ (𝐵 × 𝐵)) → ((1st ‘𝑧) + (2nd ‘𝑧)) ∈ 𝐵)
13	2, 5, 6, 12, 9	grpsubcld 33645	. . 3 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ (𝐵 × 𝐵)) → (((1st ‘𝑧) + (2nd ‘𝑧)) − (1st ‘𝑧)) ∈ 𝐵)
14		cntrval2.4	. . . 4 ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))
15		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
16		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
17	15, 16	op1std 8011	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
18	15, 16	op2ndd 8012	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
19	17, 18	oveq12d 7438	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑧) + (2nd ‘𝑧)) = (𝑥 + 𝑦))
20	19, 17	oveq12d 7438	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((1st ‘𝑧) + (2nd ‘𝑧)) − (1st ‘𝑧)) = ((𝑥 + 𝑦) − 𝑥))
21	20	mpompt 7534	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ (((1st ‘𝑧) + (2nd ‘𝑧)) − (1st ‘𝑧))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥))
22	14, 21	eqtr4i 2786	. . 3 ⊢ ⊕ = (𝑧 ∈ (𝐵 × 𝐵) ↦ (((1st ‘𝑧) + (2nd ‘𝑧)) − (1st ‘𝑧)))
23	13, 22	fmptd 7114	. 2 ⊢ (𝑀 ∈ Grp → ⊕ :(𝐵 × 𝐵)⟶𝐵)
24	14	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥)))
25		simplr 781	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → 𝑥 = (0g‘𝑀))
26		simpr 490	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧)
27	25, 26	oveq12d 7438	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → (𝑥 + 𝑦) = ((0g‘𝑀) + 𝑧))
28	27, 25	oveq12d 7438	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → ((𝑥 + 𝑦) − 𝑥) = (((0g‘𝑀) + 𝑧) − (0g‘𝑀)))
29	1	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → 𝑀 ∈ Grp)
30		eqid 2760	. . . . . . . . . . 11 ⊢ (0g‘𝑀) = (0g‘𝑀)
31	2, 30	grpidcl 19177	. . . . . . . . . 10 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝐵)
32	31	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → (0g‘𝑀) ∈ 𝐵)
33		simpllr 788	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → 𝑧 ∈ 𝐵)
34	2, 7, 29, 32, 33	grpcld 19159	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → ((0g‘𝑀) + 𝑧) ∈ 𝐵)
35	2, 30, 5	grpsubid1 19236	. . . . . . . 8 ⊢ ((𝑀 ∈ Grp ∧ ((0g‘𝑀) + 𝑧) ∈ 𝐵) → (((0g‘𝑀) + 𝑧) − (0g‘𝑀)) = ((0g‘𝑀) + 𝑧))
36	29, 34, 35	syl2anc 596	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → (((0g‘𝑀) + 𝑧) − (0g‘𝑀)) = ((0g‘𝑀) + 𝑧))
37	2, 7, 30, 29, 33	grplidd 19181	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → ((0g‘𝑀) + 𝑧) = 𝑧)
38	28, 36, 37	3eqtrd 2799	. . . . . 6 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 = (0g‘𝑀)) ∧ 𝑦 = 𝑧) → ((𝑥 + 𝑦) − 𝑥) = 𝑧)
39	38	anasss 472	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 = (0g‘𝑀) ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = 𝑧)
40	31	adantr 486	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → (0g‘𝑀) ∈ 𝐵)
41		simpr 490	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
42	24, 39, 40, 41, 41	ovmpod 7572	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ((0g‘𝑀) ⊕ 𝑧) = 𝑧)
43	1	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → 𝑀 ∈ Grp)
44		simplr 781	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ 𝐵)
45		simpr 490	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
46		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → 𝑧 ∈ 𝐵)
47	2, 7, 43, 44, 45, 46	grpassd 19157	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑢 + 𝑣) + 𝑧) = (𝑢 + (𝑣 + 𝑧)))
48	47	oveq1d 7435	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (((𝑢 + 𝑣) + 𝑧) − (𝑢 + 𝑣)) = ((𝑢 + (𝑣 + 𝑧)) − (𝑢 + 𝑣)))
49	2, 7, 43, 45, 46	grpcld 19159	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑣 + 𝑧) ∈ 𝐵)
50	2, 7, 43, 44, 49	grpcld 19159	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 + (𝑣 + 𝑧)) ∈ 𝐵)
51	2, 7, 5	grpsubsub4 19244	. . . . . . . . 9 ⊢ ((𝑀 ∈ Grp ∧ ((𝑢 + (𝑣 + 𝑧)) ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (((𝑢 + (𝑣 + 𝑧)) − 𝑣) − 𝑢) = ((𝑢 + (𝑣 + 𝑧)) − (𝑢 + 𝑣)))
52	43, 50, 45, 44, 51	syl13anc 1399	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (((𝑢 + (𝑣 + 𝑧)) − 𝑣) − 𝑢) = ((𝑢 + (𝑣 + 𝑧)) − (𝑢 + 𝑣)))
53	2, 7, 5	grpaddsubass 19241	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ (𝑣 + 𝑧) ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢 + (𝑣 + 𝑧)) − 𝑣) = (𝑢 + ((𝑣 + 𝑧) − 𝑣)))
54	43, 44, 49, 45, 53	syl13anc 1399	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑢 + (𝑣 + 𝑧)) − 𝑣) = (𝑢 + ((𝑣 + 𝑧) − 𝑣)))
55	54	oveq1d 7435	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (((𝑢 + (𝑣 + 𝑧)) − 𝑣) − 𝑢) = ((𝑢 + ((𝑣 + 𝑧) − 𝑣)) − 𝑢))
56	48, 52, 55	3eqtr2d 2801	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (((𝑢 + 𝑣) + 𝑧) − (𝑢 + 𝑣)) = ((𝑢 + ((𝑣 + 𝑧) − 𝑣)) − 𝑢))
57	14	a1i 11	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥)))
58		simprl 783	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = (𝑢 + 𝑣) ∧ 𝑦 = 𝑧)) → 𝑥 = (𝑢 + 𝑣))
59		simprr 785	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = (𝑢 + 𝑣) ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
60	58, 59	oveq12d 7438	. . . . . . . . 9 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = (𝑢 + 𝑣) ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = ((𝑢 + 𝑣) + 𝑧))
61	60, 58	oveq12d 7438	. . . . . . . 8 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = (𝑢 + 𝑣) ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = (((𝑢 + 𝑣) + 𝑧) − (𝑢 + 𝑣)))
62	2, 7, 43, 44, 45	grpcld 19159	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63		ovexd 7455	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (((𝑢 + 𝑣) + 𝑧) − (𝑢 + 𝑣)) ∈ V)
64	57, 61, 62, 46, 63	ovmpod 7572	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑢 + 𝑣) ⊕ 𝑧) = (((𝑢 + 𝑣) + 𝑧) − (𝑢 + 𝑣)))
65		simprl 783	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → 𝑥 = 𝑢)
66		simprr 785	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → 𝑦 = (𝑣 ⊕ 𝑧))
67		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑣 ∧ 𝑦 = 𝑧)) → 𝑥 = 𝑣)
68		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑣 ∧ 𝑦 = 𝑧)) → 𝑦 = 𝑧)
69	67, 68	oveq12d 7438	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑣 ∧ 𝑦 = 𝑧)) → (𝑥 + 𝑦) = (𝑣 + 𝑧))
70	69, 67	oveq12d 7438	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑣 ∧ 𝑦 = 𝑧)) → ((𝑥 + 𝑦) − 𝑥) = ((𝑣 + 𝑧) − 𝑣))
71		ovexd 7455	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑣 + 𝑧) − 𝑣) ∈ V)
72	57, 70, 45, 46, 71	ovmpod 7572	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑣 ⊕ 𝑧) = ((𝑣 + 𝑧) − 𝑣))
73	72	adantr 486	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → (𝑣 ⊕ 𝑧) = ((𝑣 + 𝑧) − 𝑣))
74	66, 73	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → 𝑦 = ((𝑣 + 𝑧) − 𝑣))
75	65, 74	oveq12d 7438	. . . . . . . . 9 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → (𝑥 + 𝑦) = (𝑢 + ((𝑣 + 𝑧) − 𝑣)))
76	75, 65	oveq12d 7438	. . . . . . . 8 ⊢ (((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ (𝑥 = 𝑢 ∧ 𝑦 = (𝑣 ⊕ 𝑧))) → ((𝑥 + 𝑦) − 𝑥) = ((𝑢 + ((𝑣 + 𝑧) − 𝑣)) − 𝑢))
77	23	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ⊕ :(𝐵 × 𝐵)⟶𝐵)
78	77, 45, 46	fovcdmd 7593	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑣 ⊕ 𝑧) ∈ 𝐵)
79		ovexd 7455	. . . . . . . 8 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑢 + ((𝑣 + 𝑧) − 𝑣)) − 𝑢) ∈ V)
80	57, 76, 44, 78, 79	ovmpod 7572	. . . . . . 7 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 ⊕ (𝑣 ⊕ 𝑧)) = ((𝑢 + ((𝑣 + 𝑧) − 𝑣)) − 𝑢))
81	56, 64, 80	3eqtr4d 2805	. . . . . 6 ⊢ ((((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
82	81	anasss 472	. . . . 5 ⊢ (((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
83	82	ralrimivva 3205	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
84	42, 83	jca 521	. . 3 ⊢ ((𝑀 ∈ Grp ∧ 𝑧 ∈ 𝐵) → (((0g‘𝑀) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
85	84	ralrimiva 3154	. 2 ⊢ (𝑀 ∈ Grp → ∀𝑧 ∈ 𝐵 (((0g‘𝑀) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))
86	2, 7, 30	isga 19506	. 2 ⊢ ( ⊕ ∈ (𝑀 GrpAct 𝐵) ↔ ((𝑀 ∈ Grp ∧ 𝐵 ∈ V) ∧ ( ⊕ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 (((0g‘𝑀) ⊕ 𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢 + 𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧))))))
87	1, 4, 23, 85, 86	syl22anbrc 33099	1 ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450  ⟨cop 4589   ↦ cmpt 5185   × cxp 5648  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17388  +_gcplusg 17429  0_gc0g 17611  Grpcgrp 19145  -_gcsg 19147   GrpAct cga 19504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8850  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-grp 19148  df-minusg 19149  df-sbg 19150  df-ga 19505

This theorem is used by:  cntrval2  33775