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Theorem gaid 19163
Description: The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaid.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
gaid ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

Proof of Theorem gaid
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3493 . . 3 (𝑆𝑉𝑆 ∈ V)
21anim2i 618 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3 gaid.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2733 . . . . . . . 8 (0g𝐺) = (0g𝐺)
53, 4grpidcl 18850 . . . . . . 7 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
65adantr 482 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (0g𝐺) ∈ 𝑋)
7 ovres 7573 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g𝐺)2nd 𝑥))
8 df-ov 7412 . . . . . . . 8 ((0g𝐺)2nd 𝑥) = (2nd ‘⟨(0g𝐺), 𝑥⟩)
9 fvex 6905 . . . . . . . . 9 (0g𝐺) ∈ V
10 vex 3479 . . . . . . . . 9 𝑥 ∈ V
119, 10op2nd 7984 . . . . . . . 8 (2nd ‘⟨(0g𝐺), 𝑥⟩) = 𝑥
128, 11eqtri 2761 . . . . . . 7 ((0g𝐺)2nd 𝑥) = 𝑥
137, 12eqtrdi 2789 . . . . . 6 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
146, 13sylan 581 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
15 simprl 770 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
16 simplr 768 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑥𝑆)
17 ovres 7573 . . . . . . . . 9 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18 df-ov 7412 . . . . . . . . . 10 (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19 vex 3479 . . . . . . . . . . 11 𝑦 ∈ V
2019, 10op2nd 7984 . . . . . . . . . 10 (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
2118, 20eqtri 2761 . . . . . . . . 9 (𝑦2nd 𝑥) = 𝑥
2217, 21eqtrdi 2789 . . . . . . . 8 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
2315, 16, 22syl2anc 585 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
24 simprr 772 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
25 ovres 7573 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26 df-ov 7412 . . . . . . . . . . 11 (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27 vex 3479 . . . . . . . . . . . 12 𝑧 ∈ V
2827, 10op2nd 7984 . . . . . . . . . . 11 (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
2926, 28eqtri 2761 . . . . . . . . . 10 (𝑧2nd 𝑥) = 𝑥
3025, 29eqtrdi 2789 . . . . . . . . 9 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3124, 16, 30syl2anc 585 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3231oveq2d 7425 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥))
33 eqid 2733 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
343, 33grpcl 18827 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
35343expb 1121 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
3635ad4ant14 751 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
37 ovres 7573 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g𝐺)𝑧)2nd 𝑥))
38 df-ov 7412 . . . . . . . . . 10 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩)
39 ovex 7442 . . . . . . . . . . 11 (𝑦(+g𝐺)𝑧) ∈ V
4039, 10op2nd 7984 . . . . . . . . . 10 (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩) = 𝑥
4138, 40eqtri 2761 . . . . . . . . 9 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = 𝑥
4237, 41eqtrdi 2789 . . . . . . . 8 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4336, 16, 42syl2anc 585 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4423, 32, 433eqtr4rd 2784 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4544ralrimivva 3201 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4614, 45jca 513 . . . 4 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → (((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))
4746ralrimiva 3147 . . 3 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → ∀𝑥𝑆 (((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))
48 f2ndres 8000 . . 3 (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
4947, 48jctil 521 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥𝑆 (((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))))
503, 33, 4isga 19155 . 2 ((2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥𝑆 (((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))))
512, 49, 50sylanbrc 584 1 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cop 4635   × cxp 5675  cres 5679  wf 6540  cfv 6544  (class class class)co 7409  2nd c2nd 7974  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819   GrpAct cga 19153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7976  df-map 8822  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-ga 19154
This theorem is referenced by: (None)
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