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Theorem grpsubpropd2 12980
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 997 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2 simp2 998 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3 grpsubpropd2.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
433ad2ant1 1018 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
52, 4eleqtrrd 2257 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎𝐵)
6 grpsubpropd2.3 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
763ad2ant1 1018 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8 simp3 999 . . . . . . . 8 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9 eqid 2177 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
10 eqid 2177 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
119, 10grpinvcl 12926 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
127, 8, 11syl2anc 411 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ (Base‘𝐺))
1312, 4eleqtrrd 2257 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑏) ∈ 𝐵)
14 grpsubpropd2.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
1514oveqrspc2v 5904 . . . . . 6 ((𝜑 ∧ (𝑎𝐵 ∧ ((invg𝐺)‘𝑏) ∈ 𝐵)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
161, 5, 13, 15syl12anc 1236 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐺)‘𝑏)))
17 grpsubpropd2.2 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐻))
18 eqid 2177 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
199, 18grpidcl 12909 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
206, 19syl 14 . . . . . . . . . . 11 (𝜑 → (0g𝐺) ∈ (Base‘𝐺))
2120, 3eleqtrrd 2257 . . . . . . . . . 10 (𝜑 → (0g𝐺) ∈ 𝐵)
2217, 21basmexd 12524 . . . . . . . . 9 (𝜑𝐻 ∈ V)
233, 17, 6, 22, 14grpinvpropdg 12950 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
2423fveq1d 5519 . . . . . . 7 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
2524oveq2d 5893 . . . . . 6 (𝜑 → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
26253ad2ant1 1018 . . . . 5 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐻)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2716, 26eqtrd 2210 . . . 4 ((𝜑𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
2827mpoeq3dva 5941 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
293, 17eqtr3d 2212 . . . 4 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
30 mpoeq12 5937 . . . 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))))
3129, 29, 30syl2anc 411 . . 3 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
3228, 31eqtrd 2210 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
33 eqid 2177 . . . 4 (+g𝐺) = (+g𝐺)
34 eqid 2177 . . . 4 (-g𝐺) = (-g𝐺)
359, 33, 10, 34grpsubfvalg 12923 . . 3 (𝐺 ∈ Grp → (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))))
366, 35syl 14 . 2 (𝜑 → (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))))
37 eqid 2177 . . . 4 (Base‘𝐻) = (Base‘𝐻)
38 eqid 2177 . . . 4 (+g𝐻) = (+g𝐻)
39 eqid 2177 . . . 4 (invg𝐻) = (invg𝐻)
40 eqid 2177 . . . 4 (-g𝐻) = (-g𝐻)
4137, 38, 39, 40grpsubfvalg 12923 . . 3 (𝐻 ∈ V → (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
4222, 41syl 14 . 2 (𝜑 → (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
4332, 36, 423eqtr4d 2220 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  Vcvv 2739  cfv 5218  (class class class)co 5877  cmpo 5879  Basecbs 12464  +gcplusg 12538  0gc0g 12710  Grpcgrp 12882  invgcminusg 12883  -gcsg 12884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-sbg 12887
This theorem is referenced by: (None)
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