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Theorem grpsubpropdg 12974
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))
grpsubpropdg.g (𝜑𝐺𝑉)
grpsubpropdg.h (𝜑𝐻𝑊)
Assertion
Ref Expression
grpsubpropdg (𝜑 → (-g𝐺) = (-g𝐻))

Proof of Theorem grpsubpropdg
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 2178 . . . 4 (𝜑𝑎 = 𝑎)
4 eqidd 2178 . . . . . 6 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5 grpsubpropdg.g . . . . . 6 (𝜑𝐺𝑉)
6 grpsubpropdg.h . . . . . 6 (𝜑𝐻𝑊)
72oveqdr 5903 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
84, 1, 5, 6, 7grpinvpropdg 12945 . . . . 5 (𝜑 → (invg𝐺) = (invg𝐻))
98fveq1d 5518 . . . 4 (𝜑 → ((invg𝐺)‘𝑏) = ((invg𝐻)‘𝑏))
102, 3, 9oveq123d 5896 . . 3 (𝜑 → (𝑎(+g𝐺)((invg𝐺)‘𝑏)) = (𝑎(+g𝐻)((invg𝐻)‘𝑏)))
111, 1, 10mpoeq123dv 5937 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
12 eqid 2177 . . . 4 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2177 . . . 4 (+g𝐺) = (+g𝐺)
14 eqid 2177 . . . 4 (invg𝐺) = (invg𝐺)
15 eqid 2177 . . . 4 (-g𝐺) = (-g𝐺)
1612, 13, 14, 15grpsubfvalg 12918 . . 3 (𝐺𝑉 → (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))))
175, 16syl 14 . 2 (𝜑 → (-g𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g𝐺)((invg𝐺)‘𝑏))))
18 eqid 2177 . . . 4 (Base‘𝐻) = (Base‘𝐻)
19 eqid 2177 . . . 4 (+g𝐻) = (+g𝐻)
20 eqid 2177 . . . 4 (invg𝐻) = (invg𝐻)
21 eqid 2177 . . . 4 (-g𝐻) = (-g𝐻)
2218, 19, 20, 21grpsubfvalg 12918 . . 3 (𝐻𝑊 → (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
236, 22syl 14 . 2 (𝜑 → (-g𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g𝐻)((invg𝐻)‘𝑏))))
2411, 17, 233eqtr4d 2220 1 (𝜑 → (-g𝐺) = (-g𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  cfv 5217  (class class class)co 5875  cmpo 5877  Basecbs 12462  +gcplusg 12536  invgcminusg 12878  -gcsg 12879
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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908
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-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468  df-0g 12707  df-minusg 12881  df-sbg 12882
This theorem is referenced by: (None)
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