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Theorem gass 19081
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 7520 . . . . 5 ((𝑥𝑋𝑦𝑍) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) = (𝑥 𝑦))
21adantl 482 . . . 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 19075 . . . . . 6 (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
54adantl 482 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
65fovcdmda 7525 . . . 4 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
72, 6eqeltrrd 2839 . . 3 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥 𝑦) ∈ 𝑍)
87ralrimivva 3197 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
9 gagrp 19072 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 724 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝐺 ∈ Grp)
11 gaset 19073 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
1211adantr 481 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑌 ∈ V)
13 simpr 485 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍𝑌)
1412, 13ssexd 5281 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍 ∈ V)
1514adantr 481 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍 ∈ V)
1610, 15jca 512 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝐺 ∈ Grp ∧ 𝑍 ∈ V))
173gaf 19075 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
1817ad2antrr 724 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → :(𝑋 × 𝑌)⟶𝑌)
1918ffnd 6669 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → Fn (𝑋 × 𝑌))
20 simplr 767 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍𝑌)
21 xpss2 5653 . . . . . . 7 (𝑍𝑌 → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
2220, 21syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
23 fnssres 6624 . . . . . 6 (( Fn (𝑋 × 𝑌) ∧ (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌)) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
2419, 22, 23syl2anc 584 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
25 simpr 485 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
261eleq1d 2822 . . . . . . . 8 ((𝑥𝑋𝑦𝑍) → ((𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ (𝑥 𝑦) ∈ 𝑍))
2726ralbidva 3172 . . . . . . 7 (𝑥𝑋 → (∀𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))
2827ralbiia 3094 . . . . . 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 7483 . . . . 5 (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ↔ (( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍))
3124, 29, 30sylanbrc 583 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
32 eqid 2736 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
333, 32grpidcl 18778 . . . . . . . . 9 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
3410, 33syl 17 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (0g𝐺) ∈ 𝑋)
35 ovres 7520 . . . . . . . 8 (((0g𝐺) ∈ 𝑋𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3634, 35sylan 580 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
37 simpll 765 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∈ (𝐺 GrpAct 𝑌))
3820sselda 3944 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → 𝑧𝑌)
3932gagrpid 19074 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4037, 38, 39syl2an2r 683 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺) 𝑧) = 𝑧)
4136, 40eqtrd 2776 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧)
4237ad2antrr 724 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∈ (𝐺 GrpAct 𝑌))
43 simprl 769 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
44 simprr 771 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4538adantr 481 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
46 eqid 2736 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
473, 46gaass 19077 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢𝑋𝑣𝑋𝑧𝑌)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
4842, 43, 44, 45, 47syl13anc 1372 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
49 simplr 767 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑍)
50 simpllr 774 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
51 ovrspc2v 7383 . . . . . . . . . . 11 (((𝑣𝑋𝑧𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑣 𝑧) ∈ 𝑍)
5244, 49, 50, 51syl21anc 836 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) ∈ 𝑍)
53 ovres 7520 . . . . . . . . . 10 ((𝑢𝑋 ∧ (𝑣 𝑧) ∈ 𝑍) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5443, 52, 53syl2anc 584 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5548, 54eqtr4d 2779 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
5610ad2antrr 724 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
573, 46grpcl 18756 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
5856, 43, 44, 57syl3anc 1371 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
59 ovres 7520 . . . . . . . . 9 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑍) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
6058, 49, 59syl2anc 584 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
61 ovres 7520 . . . . . . . . . 10 ((𝑣𝑋𝑧𝑍) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6244, 49, 61syl2anc 584 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6362oveq2d 7373 . . . . . . . 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 2786 . . . . . . 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 3197 . . . . . 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 512 . . . . 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 3143 . . . 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 512 . . 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 19071 . . 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 583 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍))
718, 70impbida 799 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 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910   × cxp 5631  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748   GrpAct cga 19069
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-ga 19070
This theorem is referenced by: (None)
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