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

Theorem eqglact 17692
Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x 𝑋 = (Base‘𝐺)
eqger.r = (𝐺 ~QG 𝑌)
eqglact.3 + = (+g𝐺)
Assertion
Ref Expression
eqglact ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Distinct variable groups:   𝑥, +   𝑥,   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
2 eqid 2651 . . . . . . 7 (invg𝐺) = (invg𝐺)
3 eqglact.3 . . . . . . 7 + = (+g𝐺)
4 eqger.r . . . . . . 7 = (𝐺 ~QG 𝑌)
51, 2, 3, 4eqgval 17690 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6 3anass 1059 . . . . . 6 ((𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
75, 6syl6bb 276 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
87baibd 968 . . . 4 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
983impa 1278 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
109abbidv 2770 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → {𝑥𝐴 𝑥} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11 dfec2 7790 . . 3 (𝐴𝑋 → [𝐴] = {𝑥𝐴 𝑥})
12113ad2ant3 1104 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = {𝑥𝐴 𝑥})
13 eqid 2651 . . . . . . . . 9 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
1413, 1, 3, 2grplactcnv 17565 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1514simprd 478 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
1613, 1grplactfval 17563 . . . . . . . . 9 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1716adantl 481 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1817cnveqd 5330 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
191, 2grpinvcl 17514 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
2013, 1grplactfval 17563 . . . . . . . 8 (((invg𝐺)‘𝐴) ∈ 𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2215, 18, 213eqtr3d 2693 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2322cnveqd 5330 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
24233adant2 1100 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2524imaeq1d 5500 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26 imacnvcnv 5634 . . 3 ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27 eqid 2651 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2827mptpreima 5666 . . . 4 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29 df-rab 2950 . . . 4 {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3028, 29eqtri 2673 . . 3 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3125, 26, 303eqtr3g 2708 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
3210, 12, 313eqtr4d 2695 1 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  {crab 2945  wss 3607   class class class wbr 4685  cmpt 4762  ccnv 5142  cima 5146  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  [cec 7785  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469  invgcminusg 17470   ~QG cqg 17637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-ec 7789  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-eqg 17640
This theorem is referenced by:  eqgen  17694  cldsubg  21961  tgpconncompeqg  21962  snclseqg  21966
  Copyright terms: Public domain W3C validator