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Theorem gass 17905
 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 6953 . . . . 5 ((𝑥𝑋𝑦𝑍) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) = (𝑥 𝑦))
21adantl 473 . . . 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 17899 . . . . . 6 (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
54adantl 473 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
65fovrnda 6958 . . . 4 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍)
72, 6eqeltrrd 2828 . . 3 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) ∧ (𝑥𝑋𝑦𝑍)) → (𝑥 𝑦) ∈ 𝑍)
87ralrimivva 3097 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
9 gagrp 17896 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 764 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝐺 ∈ Grp)
11 gaset 17897 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
1211adantr 472 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑌 ∈ V)
13 simpr 479 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍𝑌)
1412, 13ssexd 4945 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → 𝑍 ∈ V)
1514adantr 472 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍 ∈ V)
1610, 15jca 555 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝐺 ∈ Grp ∧ 𝑍 ∈ V))
173gaf 17899 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
1817ad2antrr 764 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → :(𝑋 × 𝑌)⟶𝑌)
19 ffn 6194 . . . . . . 7 ( :(𝑋 × 𝑌)⟶𝑌 Fn (𝑋 × 𝑌))
2018, 19syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → Fn (𝑋 × 𝑌))
21 simplr 809 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → 𝑍𝑌)
22 xpss2 5273 . . . . . . 7 (𝑍𝑌 → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
2321, 22syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌))
24 fnssres 6153 . . . . . 6 (( Fn (𝑋 × 𝑌) ∧ (𝑋 × 𝑍) ⊆ (𝑋 × 𝑌)) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
2520, 23, 24syl2anc 696 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍))
26 simpr 479 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
271eleq1d 2812 . . . . . . . 8 ((𝑥𝑋𝑦𝑍) → ((𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ (𝑥 𝑦) ∈ 𝑍))
2827ralbidva 3111 . . . . . . 7 (𝑥𝑋 → (∀𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍 ↔ ∀𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))
2928ralbiia 3105 . . . . . 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 6917 . . . . 5 (( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍 ↔ (( ↾ (𝑋 × 𝑍)) Fn (𝑋 × 𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥( ↾ (𝑋 × 𝑍))𝑦) ∈ 𝑍))
3225, 30, 31sylanbrc 701 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)):(𝑋 × 𝑍)⟶𝑍)
33 eqid 2748 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
343, 33grpidcl 17622 . . . . . . . . 9 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
3510, 34syl 17 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (0g𝐺) ∈ 𝑋)
36 ovres 6953 . . . . . . . 8 (((0g𝐺) ∈ 𝑋𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3735, 36sylan 489 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = ((0g𝐺) 𝑧))
3821sselda 3732 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → 𝑧𝑌)
39 simpll 807 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ∈ (𝐺 GrpAct 𝑌))
4033gagrpid 17898 . . . . . . . . 9 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4139, 40sylan 489 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4238, 41syldan 488 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺) 𝑧) = 𝑧)
4337, 42eqtrd 2782 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) → ((0g𝐺)( ↾ (𝑋 × 𝑍))𝑧) = 𝑧)
4439ad2antrr 764 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∈ (𝐺 GrpAct 𝑌))
45 simprl 811 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
46 simprr 813 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4738adantr 472 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
48 eqid 2748 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
493, 48gaass 17901 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢𝑋𝑣𝑋𝑧𝑌)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
5044, 45, 46, 47, 49syl13anc 1465 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
51 simplr 809 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑍)
52 simpllr 817 . . . . . . . . . . 11 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍)
53 ovrspc2v 6823 . . . . . . . . . . 11 (((𝑣𝑋𝑧𝑍) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → (𝑣 𝑧) ∈ 𝑍)
5446, 51, 52, 53syl21anc 1462 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) ∈ 𝑍)
55 ovres 6953 . . . . . . . . . 10 ((𝑢𝑋 ∧ (𝑣 𝑧) ∈ 𝑍) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5645, 54, 55syl2anc 696 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)) = (𝑢 (𝑣 𝑧)))
5750, 56eqtr4d 2785 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢( ↾ (𝑋 × 𝑍))(𝑣 𝑧)))
5810ad2antrr 764 . . . . . . . . . 10 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
593, 48grpcl 17602 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
6058, 45, 46, 59syl3anc 1463 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
61 ovres 6953 . . . . . . . . 9 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑍) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
6260, 51, 61syl2anc 696 . . . . . . . 8 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣)( ↾ (𝑋 × 𝑍))𝑧) = ((𝑢(+g𝐺)𝑣) 𝑧))
63 ovres 6953 . . . . . . . . . 10 ((𝑣𝑋𝑧𝑍) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6446, 51, 63syl2anc 696 . . . . . . . . 9 ((((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) ∧ 𝑧𝑍) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣( ↾ (𝑋 × 𝑍))𝑧) = (𝑣 𝑧))
6564oveq2d 6817 . . . . . . . 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 2792 . . . . . . 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 3097 . . . . . 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 555 . . . . 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 3092 . . . 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 555 . . 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 17895 . . 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 701 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) ∧ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍) → ( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍))
738, 72impbida 913 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 383   = wceq 1620   ∈ wcel 2127  ∀wral 3038  Vcvv 3328   ⊆ wss 3703   × cxp 5252   ↾ cres 5256   Fn wfn 6032  ⟶wf 6033  ‘cfv 6037  (class class class)co 6801  Basecbs 16030  +gcplusg 16114  0gc0g 16273  Grpcgrp 17594   GrpAct cga 17893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-map 8013  df-0g 16275  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-grp 17597  df-ga 17894 This theorem is referenced by: (None)
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