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Theorem grpsubpropd 17343
 Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
Hypotheses
Ref Expression
grpsubpropd.b (𝜑 → (Base‘𝐺) = (Base‘𝐻))
grpsubpropd.p (𝜑 → (+g𝐺) = (+g𝐻))
Assertion
Ref Expression
grpsubpropd (𝜑 → (-g𝐺) = (-g𝐻))

Proof of Theorem grpsubpropd
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsubpropd.b . . 3 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
2 grpsubpropd.p . . . 4 (𝜑 → (+g𝐺) = (+g𝐻))
3 eqidd 2611 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2611 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
52oveqdr 6573 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
64, 1, 5grpinvpropd 17313 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
76fveq1d 6105 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
82, 3, 7oveq123d 6570 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
91, 1, 8mpt2eq123dv 6615 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
10 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2610 . . 3 (invg𝐺) = (invg𝐺)
13 eqid 2610 . . 3 (-g𝐺) = (-g𝐺)
1410, 11, 12, 13grpsubfval 17287 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
15 eqid 2610 . . 3 (Base‘𝐻) = (Base‘𝐻)
16 eqid 2610 . . 3 (+g𝐻) = (+g𝐻)
17 eqid 2610 . . 3 (invg𝐻) = (invg𝐻)
18 eqid 2610 . . 3 (-g𝐻) = (-g𝐻)
1915, 16, 17, 18grpsubfval 17287 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
209, 14, 193eqtr4g 2669 1 (𝜑 → (-g𝐺) = (-g𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  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-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-minusg 17249  df-sbg 17250 This theorem is referenced by:  rlmsub  19019  matsubg  20057  tngngp2  22266  tngngp  22268  tchsub  22828  ply1divalg2  23702  ttgsub  25559  zhmnrg  29339
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