Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpsubpropd2 Structured version   Visualization version   GIF version

Theorem grpsubpropd2 17344
 Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubpropd2.1 (𝜑𝐵 = (Base‘𝐺))
grpsubpropd2.2 (𝜑𝐵 = (Base‘𝐻))
grpsubpropd2.3 (𝜑𝐺 ∈ Grp)
grpsubpropd2.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
Assertion
Ref Expression
grpsubpropd2 (𝜑 → (-g𝐺) = (-g𝐻))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grpsubpropd2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1054 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2 simp2 1055 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3 grpsubpropd2.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
433ad2ant1 1075 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
52, 4eleqtrrd 2691 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎𝐵)
6 grpsubpropd2.3 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
763ad2ant1 1075 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8 simp3 1056 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9 eqid 2610 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
10 eqid 2610 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 17290 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
127, 8, 11syl2anc 691 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
1312, 4eleqtrrd 2691 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ 𝐵)
14 grpsubpropd2.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
1514oveqrspc2v 6572 . . . . . 6 ((𝜑 ∧ (𝑎𝐵 ∧ ((invg𝐺)‘𝑏) ∈ 𝐵)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
161, 5, 13, 15syl12anc 1316 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
17 grpsubpropd2.2 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐻))
183, 17, 14grpinvpropd 17313 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
1918fveq1d 6105 . . . . . . 7 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
2019oveq2d 6565 . . . . . 6 (𝜑 → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
21203ad2ant1 1075 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2216, 21eqtrd 2644 . . . 4 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2322mpt2eq3dva 6617 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
243, 17eqtr3d 2646 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25 mpt2eq12 6613 . . . 4 (((Base‘𝐺) = (Base‘𝐻) ∧ (Base‘𝐺) = (Base‘𝐻)) → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2624, 24, 25syl2anc 691 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2723, 26eqtrd 2644 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
28 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
29 eqid 2610 . . 3 (-g𝐺) = (-g𝐺)
309, 28, 10, 29grpsubfval 17287 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
31 eqid 2610 . . 3 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2610 . . 3 (+g𝐻) = (+g𝐻)
33 eqid 2610 . . 3 (invg𝐻) = (invg𝐻)
34 eqid 2610 . . 3 (-g𝐻) = (-g𝐻)
3531, 32, 33, 34grpsubfval 17287 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
3627, 30, 353eqtr4g 2669 1 (𝜑 → (-g𝐺) = (-g𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247 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-1st 7059  df-2nd 7060  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250 This theorem is referenced by:  ngppropd  22251
 Copyright terms: Public domain W3C validator