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Theorem grpsubpropd2 18966
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 1135 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2 simp2 1136 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3 grpsubpropd2.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
433ad2ant1 1132 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
52, 4eleqtrrd 2835 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎𝐵)
6 grpsubpropd2.3 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
763ad2ant1 1132 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8 simp3 1137 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9 eqid 2731 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
10 eqid 2731 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 18909 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
127, 8, 11syl2anc 583 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
1312, 4eleqtrrd 2835 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ 𝐵)
14 grpsubpropd2.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
1514oveqrspc2v 7439 . . . . . 6 ((𝜑 ∧ (𝑎𝐵 ∧ ((invg𝐺)‘𝑏) ∈ 𝐵)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
161, 5, 13, 15syl12anc 834 . . . . 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 18935 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
1918fveq1d 6894 . . . . . . 7 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
2019oveq2d 7428 . . . . . 6 (𝜑 → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
21203ad2ant1 1132 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2216, 21eqtrd 2771 . . . 4 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2322mpoeq3dva 7489 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
243, 17eqtr3d 2773 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25 mpoeq12 7485 . . . 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 583 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2723, 26eqtrd 2771 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
28 eqid 2731 . . 3 (+g𝐺) = (+g𝐺)
29 eqid 2731 . . 3 (-g𝐺) = (-g𝐺)
309, 28, 10, 29grpsubfval 18905 . 2 (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏)))
31 eqid 2731 . . 3 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2731 . . 3 (+g𝐻) = (+g𝐻)
33 eqid 2731 . . 3 (invg𝐻) = (invg𝐻)
34 eqid 2731 . . 3 (-g𝐻) = (-g𝐻)
3531, 32, 33, 34grpsubfval 18905 . 2 (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
3627, 30, 353eqtr4g 2796 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  cfv 6544  (class class class)co 7412  cmpo 7414  Basecbs 17149  +gcplusg 17202  Grpcgrp 18856  invgcminusg 18857  -gcsg 18858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-sbg 18861
This theorem is referenced by:  ngppropd  24367
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