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Theorem grpsubpropd2 18925
Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubpropd2.1 (𝜑𝐵 = (Base‘𝐺))
grpsubpropd2.2 (𝜑𝐵 = (Base‘𝐻))
grpsubpropd2.3 (𝜑𝐺 ∈ Grp)
grpsubpropd2.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
Assertion
Ref Expression
grpsubpropd2 (𝜑 → (-g𝐺) = (-g𝐻))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grpsubpropd2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2 simp2 1137 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3 grpsubpropd2.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
433ad2ant1 1133 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
52, 4eleqtrrd 2836 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎𝐵)
6 grpsubpropd2.3 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
763ad2ant1 1133 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8 simp3 1138 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9 eqid 2732 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
10 eqid 2732 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 18868 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
127, 8, 11syl2anc 584 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
1312, 4eleqtrrd 2836 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ 𝐵)
14 grpsubpropd2.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
1514oveqrspc2v 7432 . . . . . 6 ((𝜑 ∧ (𝑎𝐵 ∧ ((invg𝐺)‘𝑏) ∈ 𝐵)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
161, 5, 13, 15syl12anc 835 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
17 grpsubpropd2.2 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐻))
183, 17, 14grpinvpropd 18894 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
1918fveq1d 6890 . . . . . . 7 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
2019oveq2d 7421 . . . . . 6 (𝜑 → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
21203ad2ant1 1133 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2216, 21eqtrd 2772 . . . 4 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2322mpoeq3dva 7482 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
243, 17eqtr3d 2774 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25 mpoeq12 7478 . . . 4 (((Base‘𝐺) = (Base‘𝐻) ∧ (Base‘𝐺) = (Base‘𝐻)) → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2624, 24, 25syl2anc 584 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2723, 26eqtrd 2772 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
28 eqid 2732 . . 3 (+g𝐺) = (+g𝐺)
29 eqid 2732 . . 3 (-g𝐺) = (-g𝐺)
309, 28, 10, 29grpsubfval 18864 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
31 eqid 2732 . . 3 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2732 . . 3 (+g𝐻) = (+g𝐻)
33 eqid 2732 . . 3 (invg𝐻) = (invg𝐻)
34 eqid 2732 . . 3 (-g𝐻) = (-g𝐻)
3531, 32, 33, 34grpsubfval 18864 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
3627, 30, 353eqtr4g 2797 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cfv 6540  (class class class)co 7405  cmpo 7407  Basecbs 17140  +gcplusg 17193  Grpcgrp 18815  invgcminusg 18816  -gcsg 18817
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820
This theorem is referenced by:  ngppropd  24137
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