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Theorem gass 17674
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 6765 . . . . 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 17668 . . . . . 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)
65fovrnda 6770 . . . 4 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
72, 6eqeltrrd 2699 . . 3 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥 𝑦) ∈ 𝑍)
87ralrimivva 2967 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
9 gagrp 17665 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 761 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝐺 ∈ Grp)
11 gaset 17666 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
1211adantr 481 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑌 ∈ V)
13 simpr 477 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍𝑌)
1412, 13ssexd 4775 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍 ∈ V)
1514adantr 481 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍 ∈ V)
1610, 15jca 554 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝐺 ∈ Grp ∧ 𝑍 ∈ V))
173gaf 17668 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
1817ad2antrr 761 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → :(𝑋 × 𝑌)⟶𝑌)
19 ffn 6012 . . . . . . 7 ( :(𝑋 × 𝑌)⟶𝑌 Fn (𝑋 × 𝑌))
2018, 19syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → Fn (𝑋 × 𝑌))
21 simplr 791 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍𝑌)
22 xpss2 5200 . . . . . . 7 (𝑍𝑌 → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
2321, 22syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
24 fnssres 5972 . . . . . 6 (( Fn (𝑋 × 𝑌) ∧ (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌)) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
2520, 23, 24syl2anc 692 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
26 simpr 477 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
271eleq1d 2683 . . . . . . . 8 ((𝑥𝑋𝑦𝑍) → ((𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ (𝑥 𝑦) ∈ 𝑍))
2827ralbidva 2981 . . . . . . 7 (𝑥𝑋 → (∀𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))
2928ralbiia 2975 . . . . . 6 (∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
3026, 29sylibr 224 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
31 ffnov 6729 . . . . 5 (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ↔ (( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍))
3225, 30, 31sylanbrc 697 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
33 eqid 2621 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
343, 33grpidcl 17390 . . . . . . . . 9 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
3510, 34syl 17 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (0g𝐺) ∈ 𝑋)
36 ovres 6765 . . . . . . . 8 (((0g𝐺) ∈ 𝑋𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3735, 36sylan 488 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3821sselda 3588 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → 𝑧𝑌)
39 simpll 789 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∈ (𝐺 GrpAct 𝑌))
4033gagrpid 17667 . . . . . . . . 9 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4139, 40sylan 488 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4238, 41syldan 487 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺) 𝑧) = 𝑧)
4337, 42eqtrd 2655 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧)
4439ad2antrr 761 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∈ (𝐺 GrpAct 𝑌))
45 simprl 793 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
46 simprr 795 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4738adantr 481 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
48 eqid 2621 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
493, 48gaass 17670 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢𝑋𝑣𝑋𝑧𝑌)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
5044, 45, 46, 47, 49syl13anc 1325 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
51 simplr 791 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑍)
52 simpllr 798 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
53 ovrspc2v 6637 . . . . . . . . . . 11 (((𝑣𝑋𝑧𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑣 𝑧) ∈ 𝑍)
5446, 51, 52, 53syl21anc 1322 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) ∈ 𝑍)
55 ovres 6765 . . . . . . . . . 10 ((𝑢𝑋 ∧ (𝑣 𝑧) ∈ 𝑍) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5645, 54, 55syl2anc 692 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5750, 56eqtr4d 2658 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
5810ad2antrr 761 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
593, 48grpcl 17370 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
6058, 45, 46, 59syl3anc 1323 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
61 ovres 6765 . . . . . . . . 9 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑍) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
6260, 51, 61syl2anc 692 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
63 ovres 6765 . . . . . . . . . 10 ((𝑣𝑋𝑧𝑍) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6446, 51, 63syl2anc 692 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6564oveq2d 6631 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
6657, 62, 653eqtr4d 2665 . . . . . . 7 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)))
6766ralrimivva 2967 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣( ↾ (𝑋 × 𝑍))𝑧)))
6843, 67jca 554 . . . . 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))))
6968ralrimiva 2962 . . . 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))))
7032, 69jca 554 . . 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)))))
713, 48, 33isga 17664 . . 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))))))
7216, 70, 71sylanbrc 697 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍))
738, 72impbida 876 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 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  wss 3560   × cxp 5082  cres 5086   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Grpcgrp 17362   GrpAct cga 17662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-ga 17663
This theorem is referenced by: (None)
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