Theorem gass 19245

Description: A subset of a group action is a group action iff it is closed under the group action operation. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypothesis

Ref	Expression
gass.1	⊢ 𝑋 = (Base‘𝐺)

Assertion

Ref	Expression
gass	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦   𝑥, ⊕ ,𝑦

Proof of Theorem gass

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

Step	Hyp	Ref	Expression
1		ovres 7581	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑍) → (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) = (𝑥 ⊕ 𝑦))
2	1	adantl 481	. . . 4 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑍)) → (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) = (𝑥 ⊕ 𝑦))
3		gass.1	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
4	3	gaf 19239	. . . . . 6 ⊢ (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) → ( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
5	4	adantl 481	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
6	5	fovcdmda 7586	. . . 4 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑍)) → (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
7	2, 6	eqeltrrd 2830	. . 3 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑍)) → (𝑥 ⊕ 𝑦) ∈ 𝑍)
8	7	ralrimivva 3196	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍)
9		gagrp 19236	. . . . 5 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
10	9	ad2antrr 725	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → 𝐺 ∈ Grp)
11		gaset 19237	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
12	11	adantr 480	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → 𝑌 ∈ V)
13		simpr 484	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → 𝑍 ⊆ 𝑌)
14	12, 13	ssexd 5318	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ V)
15	14	adantr 480	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → 𝑍 ∈ V)
16	10, 15	jca 511	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → (𝐺 ∈ Grp ∧ 𝑍 ∈ V))
17	3	gaf 19239	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
18	17	ad2antrr 725	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
19	18	ffnd 6717	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ⊕ Fn (𝑋 × 𝑌))
20		simplr 768	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → 𝑍 ⊆ 𝑌)
21		xpss2 5692	. . . . . . 7 ⊢ (𝑍 ⊆ 𝑌 → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
22	20, 21	syl 17	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
23		fnssres 6672	. . . . . 6 ⊢ (( ⊕ Fn (𝑋 × 𝑌) ∧ (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌)) → ( ⊕ ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
24	19, 22, 23	syl2anc 583	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ( ⊕ ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
25		simpr 484	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍)
26	1	eleq1d 2814	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑍) → ((𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ (𝑥 ⊕ 𝑦) ∈ 𝑍))
27	26	ralbidva 3171	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (∀𝑦 ∈ 𝑍 (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍))
28	27	ralbiia 3087	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍)
29	25, 28	sylibr 233	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
30		ffnov 7541	. . . . 5 ⊢ (( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ↔ (( ⊕ ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥( ⊕ ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍))
31	24, 29, 30	sylanbrc 582	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
32		eqid 2728	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
33	3, 32	grpidcl 18915	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
34	10, 33	syl 17	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → (0g‘𝐺) ∈ 𝑋)
35		ovres 7581	. . . . . . . 8 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑧 ∈ 𝑍) → ((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = ((0g‘𝐺) ⊕ 𝑧))
36	34, 35	sylan 579	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → ((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = ((0g‘𝐺) ⊕ 𝑧))
37		simpll 766	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ⊕ ∈ (𝐺 GrpAct 𝑌))
38	20	sselda 3978	. . . . . . . 8 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ 𝑌)
39	32	gagrpid 19238	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑧 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝑧) = 𝑧)
40	37, 38, 39	syl2an2r 684	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → ((0g‘𝐺) ⊕ 𝑧) = 𝑧)
41	36, 40	eqtrd 2768	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → ((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = 𝑧)
42	37	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
43		simprl 770	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑢 ∈ 𝑋)
44		simprr 772	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑣 ∈ 𝑋)
45	38	adantr 480	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑧 ∈ 𝑌)
46		eqid 2728	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
47	3, 46	gaass 19241	. . . . . . . . . 10 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑌)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
48	42, 43, 44, 45, 47	syl13anc 1370	. . . . . . . . 9 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
49		simplr 768	. . . . . . . . . . 11 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝑧 ∈ 𝑍)
50		simpllr 775	. . . . . . . . . . 11 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍)
51		ovrspc2v 7440	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑍) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → (𝑣 ⊕ 𝑧) ∈ 𝑍)
52	44, 49, 50, 51	syl21anc 837	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ⊕ 𝑧) ∈ 𝑍)
53		ovres 7581	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑣 ⊕ 𝑧) ∈ 𝑍) → (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣 ⊕ 𝑧)) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
54	43, 52, 53	syl2anc 583	. . . . . . . . 9 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣 ⊕ 𝑧)) = (𝑢 ⊕ (𝑣 ⊕ 𝑧)))
55	48, 54	eqtr4d 2771	. . . . . . . 8 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣 ⊕ 𝑧)))
56	10	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → 𝐺 ∈ Grp)
57	3, 46	grpcl 18891	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
58	56, 43, 44, 57	syl3anc 1369	. . . . . . . . 9 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢(+g‘𝐺)𝑣) ∈ 𝑋)
59		ovres 7581	. . . . . . . . 9 ⊢ (((𝑢(+g‘𝐺)𝑣) ∈ 𝑋 ∧ 𝑧 ∈ 𝑍) → ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧))
60	58, 49, 59	syl2anc 583	. . . . . . . 8 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g‘𝐺)𝑣) ⊕ 𝑧))
61		ovres 7581	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑋 ∧ 𝑧 ∈ 𝑍) → (𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑣 ⊕ 𝑧))
62	44, 49, 61	syl2anc 583	. . . . . . . . 9 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑣 ⊕ 𝑧))
63	62	oveq2d 7430	. . . . . . . 8 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧)) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣 ⊕ 𝑧)))
64	55, 60, 63	3eqtr4d 2778	. . . . . . 7 ⊢ ((((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧)))
65	64	ralrimivva 3196	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧)))
66	41, 65	jca 511	. . . . 5 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) ∧ 𝑧 ∈ 𝑍) → (((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧))))
67	66	ralrimiva 3142	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ∀𝑧 ∈ 𝑍 (((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧))))
68	31, 67	jca 511	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → (( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ∧ ∀𝑧 ∈ 𝑍 (((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧)))))
69	3, 46, 32	isga 19235	. . 3 ⊢ (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ((𝐺 ∈ Grp ∧ 𝑍 ∈ V) ∧ (( ⊕ ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ∧ ∀𝑧 ∈ 𝑍 (((0g‘𝐺)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 ((𝑢(+g‘𝐺)𝑣)( ⊕ ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ⊕ ↾ (𝑋 × 𝑍))(𝑣( ⊕ ↾ (𝑋 × 𝑍))𝑧))))))
70	16, 68, 69	sylanbrc 582	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍) → ( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍))
71	8, 70	impbida 800	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍 ⊆ 𝑌) → (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3057  Vcvv 3470   ⊆ wss 3945   × cxp 5670   ↾ cres 5674   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  Basecbs 17173  +_gcplusg 17226  0_gc0g 17414  Grpcgrp 18883   GrpAct cga 19233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-0g 17416  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-grp 18886  df-ga 19234

This theorem is referenced by: (None)