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Theorem grpsubpropd 17285
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 2606 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2606 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
52oveqdr 6547 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
64, 1, 5grpinvpropd 17255 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
76fveq1d 6086 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
82, 3, 7oveq123d 6544 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
91, 1, 8mpt2eq123dv 6589 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
10 eqid 2605 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2605 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2605 . . 3 (invg𝐺) = (invg𝐺)
13 eqid 2605 . . 3 (-g𝐺) = (-g𝐺)
1410, 11, 12, 13grpsubfval 17229 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
15 eqid 2605 . . 3 (Base‘𝐻) = (Base‘𝐻)
16 eqid 2605 . . 3 (+g𝐻) = (+g𝐻)
17 eqid 2605 . . 3 (invg𝐻) = (invg𝐻)
18 eqid 2605 . . 3 (-g𝐻) = (-g𝐻)
1915, 16, 17, 18grpsubfval 17229 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
209, 14, 193eqtr4g 2664 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  cfv 5786  (class class class)co 6523  cmpt2 6525  Basecbs 15637  +gcplusg 15710  invgcminusg 17188  -gcsg 17189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-0g 15867  df-minusg 17191  df-sbg 17192
This theorem is referenced by:  rlmsub  18961  matsubg  19995  tngngp2  22200  tngngp  22202  tchsub  22745  ply1divalg2  23615  ttgsub  25473  zhmnrg  29141
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