Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  gapm Structured version   Visualization version   GIF version

Theorem gapm 17562
 Description: The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gapm.1 𝑋 = (Base‘𝐺)
gapm.2 𝐹 = (𝑥𝑌 ↦ (𝐴 𝑥))
Assertion
Ref Expression
gapm (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem gapm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 gapm.2 . 2 𝐹 = (𝑥𝑌 ↦ (𝐴 𝑥))
2 gapm.1 . . . . 5 𝑋 = (Base‘𝐺)
32gaf 17551 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
43ad2antrr 758 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → :(𝑋 × 𝑌)⟶𝑌)
5 simplr 788 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝐴𝑋)
6 simpr 476 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → 𝑥𝑌)
74, 5, 6fovrnd 6704 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑥𝑌) → (𝐴 𝑥) ∈ 𝑌)
83ad2antrr 758 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → :(𝑋 × 𝑌)⟶𝑌)
9 gagrp 17548 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
109ad2antrr 758 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐺 ∈ Grp)
11 simplr 788 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝐴𝑋)
12 eqid 2610 . . . . 5 (invg𝐺) = (invg𝐺)
132, 12grpinvcl 17290 . . . 4 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
1410, 11, 13syl2anc 691 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → ((invg𝐺)‘𝐴) ∈ 𝑋)
15 simpr 476 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
168, 14, 15fovrnd 6704 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → (((invg𝐺)‘𝐴) 𝑦) ∈ 𝑌)
17 simpll 786 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ∈ (𝐺 GrpAct 𝑌))
18 simplr 788 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝐴𝑋)
19 simprl 790 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
20 simprr 792 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
212, 12gacan 17561 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2217, 18, 19, 20, 21syl13anc 1320 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐴 𝑥) = 𝑦 ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥))
2322bicomd 212 . . 3 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → ((((invg𝐺)‘𝐴) 𝑦) = 𝑥 ↔ (𝐴 𝑥) = 𝑦))
24 eqcom 2617 . . 3 (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ (((invg𝐺)‘𝐴) 𝑦) = 𝑥)
25 eqcom 2617 . . 3 (𝑦 = (𝐴 𝑥) ↔ (𝐴 𝑥) = 𝑦)
2623, 24, 253bitr4g 302 . 2 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥 = (((invg𝐺)‘𝐴) 𝑦) ↔ 𝑦 = (𝐴 𝑥)))
271, 7, 16, 26f1o2d 6785 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Grpcgrp 17245  invgcminusg 17246   GrpAct cga 17545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-ga 17546 This theorem is referenced by:  galactghm  17646
 Copyright terms: Public domain W3C validator