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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) ∧ 𝑍𝑌) → (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥𝑋𝑦𝑍 (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.
StepHypRef Expression
1 ovres 7581 . . . . 5 ((𝑥𝑋𝑦𝑍) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) = (𝑥 𝑦))
21adantl 481 . . . 4 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) = (𝑥 𝑦))
3 gass.1 . . . . . . 7 𝑋 = (Base‘𝐺)
43gaf 19239 . . . . . 6 (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
54adantl 481 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
65fovcdmda 7586 . . . 4 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
72, 6eqeltrrd 2830 . . 3 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥 𝑦) ∈ 𝑍)
87ralrimivva 3196 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
9 gagrp 19236 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 725 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝐺 ∈ Grp)
11 gaset 19237 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
1211adantr 480 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑌 ∈ V)
13 simpr 484 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍𝑌)
1412, 13ssexd 5318 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍 ∈ V)
1514adantr 480 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍 ∈ V)
1610, 15jca 511 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝐺 ∈ Grp ∧ 𝑍 ∈ V))
173gaf 19239 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
1817ad2antrr 725 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → :(𝑋 × 𝑌)⟶𝑌)
1918ffnd 6717 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → Fn (𝑋 × 𝑌))
20 simplr 768 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍𝑌)
21 xpss2 5692 . . . . . . 7 (𝑍𝑌 → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
2220, 21syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
23 fnssres 6672 . . . . . 6 (( Fn (𝑋 × 𝑌) ∧ (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌)) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
2419, 22, 23syl2anc 583 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
25 simpr 484 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
261eleq1d 2814 . . . . . . . 8 ((𝑥𝑋𝑦𝑍) → ((𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ (𝑥 𝑦) ∈ 𝑍))
2726ralbidva 3171 . . . . . . 7 (𝑥𝑋 → (∀𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))
2827ralbiia 3087 . . . . . 6 (∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
2925, 28sylibr 233 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
30 ffnov 7541 . . . . 5 (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ↔ (( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍))
3124, 29, 30sylanbrc 582 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
32 eqid 2728 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
333, 32grpidcl 18915 . . . . . . . . 9 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
3410, 33syl 17 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (0g𝐺) ∈ 𝑋)
35 ovres 7581 . . . . . . . 8 (((0g𝐺) ∈ 𝑋𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3634, 35sylan 579 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
37 simpll 766 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∈ (𝐺 GrpAct 𝑌))
3820sselda 3978 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → 𝑧𝑌)
3932gagrpid 19238 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4037, 38, 39syl2an2r 684 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺) 𝑧) = 𝑧)
4136, 40eqtrd 2768 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧)
4237ad2antrr 725 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∈ (𝐺 GrpAct 𝑌))
43 simprl 770 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
44 simprr 772 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4538adantr 480 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
46 eqid 2728 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
473, 46gaass 19241 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢𝑋𝑣𝑋𝑧𝑌)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
4842, 43, 44, 45, 47syl13anc 1370 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
49 simplr 768 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑍)
50 simpllr 775 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
51 ovrspc2v 7440 . . . . . . . . . . 11 (((𝑣𝑋𝑧𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑣 𝑧) ∈ 𝑍)
5244, 49, 50, 51syl21anc 837 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) ∈ 𝑍)
53 ovres 7581 . . . . . . . . . 10 ((𝑢𝑋 ∧ (𝑣 𝑧) ∈ 𝑍) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5443, 52, 53syl2anc 583 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5548, 54eqtr4d 2771 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
5610ad2antrr 725 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
573, 46grpcl 18891 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
5856, 43, 44, 57syl3anc 1369 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
59 ovres 7581 . . . . . . . . 9 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑍) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
6058, 49, 59syl2anc 583 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
61 ovres 7581 . . . . . . . . . 10 ((𝑣𝑋𝑧𝑍) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6244, 49, 61syl2anc 583 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6362oveq2d 7430 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
6455, 60, 633eqtr4d 2778 . . . . . . 7 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)))
6564ralrimivva 3196 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)))
6641, 65jca 511 . . . . 5 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → (((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧))))
6766ralrimiva 3142 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑧𝑍 (((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧))))
6831, 67jca 511 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ∧ ∀𝑧𝑍 (((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)))))
693, 46, 32isga 19235 . . 3 (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ((𝐺 ∈ Grp ∧ 𝑍 ∈ V) ∧ (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ∧ ∀𝑧𝑍 (((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧))))))
7016, 68, 69sylanbrc 582 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍))
718, 70impbida 800 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))
Colors of variables: wff setvar class
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  0gc0g 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)
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