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

Theorem grpsubpropd 17728
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 2772 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2772 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
52oveqdr 6819 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
64, 1, 5grpinvpropd 17698 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
76fveq1d 6334 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
82, 3, 7oveq123d 6814 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
91, 1, 8mpt2eq123dv 6864 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
10 eqid 2771 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2771 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2771 . . 3 (invg𝐺) = (invg𝐺)
13 eqid 2771 . . 3 (-g𝐺) = (-g𝐺)
1410, 11, 12, 13grpsubfval 17672 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
15 eqid 2771 . . 3 (Base‘𝐻) = (Base‘𝐻)
16 eqid 2771 . . 3 (+g𝐻) = (+g𝐻)
17 eqid 2771 . . 3 (invg𝐻) = (invg𝐻)
18 eqid 2771 . . 3 (-g𝐻) = (-g𝐻)
1915, 16, 17, 18grpsubfval 17672 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
209, 14, 193eqtr4g 2830 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  cmpt2 6795  Basecbs 16064  +gcplusg 16149  invgcminusg 17631  -gcsg 17632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-0g 16310  df-minusg 17634  df-sbg 17635
This theorem is referenced by:  rlmsub  19413  matsubg  20455  tngngp2  22676  tngngp  22678  tchsub  23239  ply1divalg2  24118  ttgsub  25980  zhmnrg  30351
  Copyright terms: Public domain W3C validator