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Theorem grpsubpropd 18286
 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 2760 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2760 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
52oveqdr 7185 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
64, 1, 5grpinvpropd 18256 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
76fveq1d 6666 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
82, 3, 7oveq123d 7178 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
91, 1, 8mpoeq123dv 7230 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
10 eqid 2759 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2759 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2759 . . 3 (invg𝐺) = (invg𝐺)
13 eqid 2759 . . 3 (-g𝐺) = (-g𝐺)
1410, 11, 12, 13grpsubfval 18229 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
15 eqid 2759 . . 3 (Base‘𝐻) = (Base‘𝐻)
16 eqid 2759 . . 3 (+g𝐻) = (+g𝐻)
17 eqid 2759 . . 3 (invg𝐻) = (invg𝐻)
18 eqid 2759 . . 3 (-g𝐻) = (-g𝐻)
1915, 16, 17, 18grpsubfval 18229 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
209, 14, 193eqtr4g 2819 1 (𝜑 → (-g𝐺) = (-g𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ‘cfv 6341  (class class class)co 7157   ∈ cmpo 7159  Basecbs 16556  +gcplusg 16638  invgcminusg 18185  -gcsg 18186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-0g 16788  df-minusg 18188  df-sbg 18189 This theorem is referenced by:  rlmsub  20053  matsubg  21147  tngngp2  23369  tngngp  23371  tcphsub  23936  ply1divalg2  24853  ttgsub  26787  zhmnrg  31450
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