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Theorem gaid 18367
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 3510 . . 3 (𝑆𝑉𝑆 ∈ V)
21anim2i 616 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3 gaid.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2818 . . . . . . . 8 (0g𝐺) = (0g𝐺)
53, 4grpidcl 18069 . . . . . . 7 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
65adantr 481 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (0g𝐺) ∈ 𝑋)
7 ovres 7303 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g𝐺)2nd 𝑥))
8 df-ov 7148 . . . . . . . 8 ((0g𝐺)2nd 𝑥) = (2nd ‘⟨(0g𝐺), 𝑥⟩)
9 fvex 6676 . . . . . . . . 9 (0g𝐺) ∈ V
10 vex 3495 . . . . . . . . 9 𝑥 ∈ V
119, 10op2nd 7687 . . . . . . . 8 (2nd ‘⟨(0g𝐺), 𝑥⟩) = 𝑥
128, 11eqtri 2841 . . . . . . 7 ((0g𝐺)2nd 𝑥) = 𝑥
137, 12syl6eq 2869 . . . . . 6 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
146, 13sylan 580 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
15 simprl 767 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
16 simplr 765 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑥𝑆)
17 ovres 7303 . . . . . . . . 9 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18 df-ov 7148 . . . . . . . . . 10 (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
2019, 10op2nd 7687 . . . . . . . . . 10 (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
2118, 20eqtri 2841 . . . . . . . . 9 (𝑦2nd 𝑥) = 𝑥
2217, 21syl6eq 2869 . . . . . . . 8 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
2315, 16, 22syl2anc 584 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
24 simprr 769 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
25 ovres 7303 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26 df-ov 7148 . . . . . . . . . . 11 (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27 vex 3495 . . . . . . . . . . . 12 𝑧 ∈ V
2827, 10op2nd 7687 . . . . . . . . . . 11 (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
2926, 28eqtri 2841 . . . . . . . . . 10 (𝑧2nd 𝑥) = 𝑥
3025, 29syl6eq 2869 . . . . . . . . 9 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3124, 16, 30syl2anc 584 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
3231oveq2d 7161 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥))
33 eqid 2818 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
343, 33grpcl 18049 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
35343expb 1112 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
3635ad4ant14 748 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
37 ovres 7303 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g𝐺)𝑧)2nd 𝑥))
38 df-ov 7148 . . . . . . . . . 10 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩)
39 ovex 7178 . . . . . . . . . . 11 (𝑦(+g𝐺)𝑧) ∈ V
4039, 10op2nd 7687 . . . . . . . . . 10 (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩) = 𝑥
4138, 40eqtri 2841 . . . . . . . . 9 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = 𝑥
4237, 41syl6eq 2869 . . . . . . . 8 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4336, 16, 42syl2anc 584 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
4423, 32, 433eqtr4rd 2864 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4544ralrimivva 3188 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4614, 45jca 512 . . . 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 3179 . . 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 7703 . . 3 (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
4947, 48jctil 520 . 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 18359 . 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 583 1 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cop 4563   × cxp 5546  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  2nd c2nd 7677  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041   GrpAct cga 18357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-2nd 7679  df-map 8397  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-ga 18358
This theorem is referenced by: (None)
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