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Theorem gaid 19317
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 3501 . . 3 (𝑆𝑉𝑆 ∈ V)
21anim2i 617 . 2 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3 gaid.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2737 . . . . . . . 8 (0g𝐺) = (0g𝐺)
53, 4grpidcl 18983 . . . . . . 7 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
65adantr 480 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (0g𝐺) ∈ 𝑋)
7 ovres 7599 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑥𝑆) → ((0g𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g𝐺)2nd 𝑥))
8 df-ov 7434 . . . . . . . 8 ((0g𝐺)2nd 𝑥) = (2nd ‘⟨(0g𝐺), 𝑥⟩)
9 fvex 6919 . . . . . . . . 9 (0g𝐺) ∈ V
10 vex 3484 . . . . . . . . 9 𝑥 ∈ V
119, 10op2nd 8023 . . . . . . . 8 (2nd ‘⟨(0g𝐺), 𝑥⟩) = 𝑥
128, 11eqtri 2765 . . . . . . 7 ((0g𝐺)2nd 𝑥) = 𝑥
137, 12eqtrdi 2793 . . . . . 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 771 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
16 simplr 769 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑥𝑆)
17 ovres 7599 . . . . . . . . 9 ((𝑦𝑋𝑥𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18 df-ov 7434 . . . . . . . . . 10 (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19 vex 3484 . . . . . . . . . . 11 𝑦 ∈ V
2019, 10op2nd 8023 . . . . . . . . . 10 (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
2118, 20eqtri 2765 . . . . . . . . 9 (𝑦2nd 𝑥) = 𝑥
2217, 21eqtrdi 2793 . . . . . . . 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 773 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
25 ovres 7599 . . . . . . . . . 10 ((𝑧𝑋𝑥𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26 df-ov 7434 . . . . . . . . . . 11 (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27 vex 3484 . . . . . . . . . . . 12 𝑧 ∈ V
2827, 10op2nd 8023 . . . . . . . . . . 11 (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
2926, 28eqtri 2765 . . . . . . . . . 10 (𝑧2nd 𝑥) = 𝑥
3025, 29eqtrdi 2793 . . . . . . . . 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 7447 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥))
33 eqid 2737 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
343, 33grpcl 18959 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
35343expb 1121 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
3635ad4ant14 752 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(+g𝐺)𝑧) ∈ 𝑋)
37 ovres 7599 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑋𝑥𝑆) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g𝐺)𝑧)2nd 𝑥))
38 df-ov 7434 . . . . . . . . . 10 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩)
39 ovex 7464 . . . . . . . . . . 11 (𝑦(+g𝐺)𝑧) ∈ V
4039, 10op2nd 8023 . . . . . . . . . 10 (2nd ‘⟨(𝑦(+g𝐺)𝑧), 𝑥⟩) = 𝑥
4138, 40eqtri 2765 . . . . . . . . 9 ((𝑦(+g𝐺)𝑧)2nd 𝑥) = 𝑥
4237, 41eqtrdi 2793 . . . . . . . 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 2788 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4544ralrimivva 3202 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑆𝑉) ∧ 𝑥𝑆) → ∀𝑦𝑋𝑧𝑋 ((𝑦(+g𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
4614, 45jca 511 . . . 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 3146 . . 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 8039 . . 3 (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
4947, 48jctil 519 . 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 19309 . 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 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cop 4632   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  2nd c2nd 8013  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951   GrpAct cga 19307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ga 19308
This theorem is referenced by: (None)
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