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Theorem gaid 17653
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 3198 . . 3 (𝑆𝑉𝑆 ∈ V)
21anim2i 592 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3 gaid.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2621 . . . . . . . 8 (0g𝐺) = (0g𝐺)
53, 4grpidcl 17371 . . . . . . 7 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
65adantr 481 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (0g𝐺) ∈ 𝑋)
7 ovres 6753 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g𝐺)2nd 𝑥))
8 df-ov 6607 . . . . . . . 8 ((0g𝐺)2nd 𝑥) = (2nd ‘⟨(0g𝐺), 𝑥⟩)
9 fvex 6158 . . . . . . . . 9 (0g𝐺) ∈ V
10 vex 3189 . . . . . . . . 9 𝑥 ∈ V
119, 10op2nd 7122 . . . . . . . 8 (2nd ‘⟨(0g𝐺), 𝑥⟩) = 𝑥
128, 11eqtri 2643 . . . . . . 7 ((0g𝐺)2nd 𝑥) = 𝑥
137, 12syl6eq 2671 . . . . . 6 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
146, 13sylan 488 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
15 simprl 793 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
16 simplr 791 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑥𝑆)
17 ovres 6753 . . . . . . . . 9 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18 df-ov 6607 . . . . . . . . . 10 (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19 vex 3189 . . . . . . . . . . 11 𝑦 ∈ V
2019, 10op2nd 7122 . . . . . . . . . 10 (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
2118, 20eqtri 2643 . . . . . . . . 9 (𝑦2nd 𝑥) = 𝑥
2217, 21syl6eq 2671 . . . . . . . 8 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
2315, 16, 22syl2anc 692 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
24 simprr 795 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
25 ovres 6753 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26 df-ov 6607 . . . . . . . . . . 11 (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27 vex 3189 . . . . . . . . . . . 12 𝑧 ∈ V
2827, 10op2nd 7122 . . . . . . . . . . 11 (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
2926, 28eqtri 2643 . . . . . . . . . 10 (𝑧2nd 𝑥) = 𝑥
3025, 29syl6eq 2671 . . . . . . . . 9 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3124, 16, 30syl2anc 692 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3231oveq2d 6620 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥))
33 simpll 789 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → 𝐺 ∈ Grp)
34 eqid 2621 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
353, 34grpcl 17351 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
36353expb 1263 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
3733, 36sylan 488 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
38 ovres 6753 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g𝐺)𝑧)2nd 𝑥))
39 df-ov 6607 . . . . . . . . . 10 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩)
40 ovex 6632 . . . . . . . . . . 11 (𝑦(+g𝐺)𝑧) ∈ V
4140, 10op2nd 7122 . . . . . . . . . 10 (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩) = 𝑥
4239, 41eqtri 2643 . . . . . . . . 9 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = 𝑥
4338, 42syl6eq 2671 . . . . . . . 8 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4437, 16, 43syl2anc 692 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4523, 32, 443eqtr4rd 2666 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4645ralrimivva 2965 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4714, 46jca 554 . . . 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))))
4847ralrimiva 2960 . . 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))))
49 f2ndres 7136 . . 3 (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
5048, 49jctil 559 . 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)))))
513, 34, 4isga 17645 . 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))))))
522, 50, 51sylanbrc 697 1 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cop 4154   × cxp 5072  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  2nd c2nd 7112  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343   GrpAct cga 17643
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-2nd 7114  df-map 7804  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-ga 17644
This theorem is referenced by: (None)
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