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Theorem grpsubpropd 18811
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 2738 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2738 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
52oveqdr 7379 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
64, 1, 5grpinvpropd 18781 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
76fveq1d 6841 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
82, 3, 7oveq123d 7372 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
91, 1, 8mpoeq123dv 7426 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
10 eqid 2737 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2737 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2737 . . 3 (invg𝐺) = (invg𝐺)
13 eqid 2737 . . 3 (-g𝐺) = (-g𝐺)
1410, 11, 12, 13grpsubfval 18754 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
15 eqid 2737 . . 3 (Base‘𝐻) = (Base‘𝐻)
16 eqid 2737 . . 3 (+g𝐻) = (+g𝐻)
17 eqid 2737 . . 3 (invg𝐻) = (invg𝐻)
18 eqid 2737 . . 3 (-g𝐻) = (-g𝐻)
1915, 16, 17, 18grpsubfval 18754 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
209, 14, 193eqtr4g 2802 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17043  +gcplusg 17093  invgcminusg 18709  -gcsg 18710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-0g 17283  df-minusg 18712  df-sbg 18713
This theorem is referenced by:  rlmsub  20620  matsubg  21733  tngngp2  23968  tngngp  23970  tcphsub  24537  ply1divalg2  25455  ttgsub  27654  zhmnrg  32360
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