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Theorem grpsubfval 17908
Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5088, see grpsubfvalALT 17909. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5088. (Revised by Rohan Ridenour, 17-Aug-2023.)
Hypotheses
Ref Expression
grpsubval.b 𝐵 = (Base‘𝐺)
grpsubval.p + = (+g𝐺)
grpsubval.i 𝐼 = (invg𝐺)
grpsubval.m = (-g𝐺)
Assertion
Ref Expression
grpsubfval = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem grpsubfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpsubval.m . . 3 = (-g𝐺)
2 fveq2 6545 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2851 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6545 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6syl6eqr 2851 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2798 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6545 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10syl6eqr 2851 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6547 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 7044 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpoeq123dv 7094 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 17870 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
163fvexi 6559 . . . . 5 𝐵 ∈ V
176fvexi 6559 . . . . . . 7 + ∈ V
1817rnex 7480 . . . . . 6 ran + ∈ V
19 p0ex 5182 . . . . . 6 {∅} ∈ V
2018, 19unex 7333 . . . . 5 (ran + ∪ {∅}) ∈ V
21 df-ov 7026 . . . . . . 7 (𝑥 + (𝐼𝑦)) = ( + ‘⟨𝑥, (𝐼𝑦)⟩)
22 fvrn0 6573 . . . . . . 7 ( + ‘⟨𝑥, (𝐼𝑦)⟩) ∈ (ran + ∪ {∅})
2321, 22eqeltri 2881 . . . . . 6 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2423rgen2w 3120 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2516, 16, 20, 24mpoexw 7639 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
2614, 15, 25fvmpt 6642 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
271, 26syl5eq 2845 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
28 fvprc 6538 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
291, 28syl5eq 2845 . . 3 𝐺 ∈ V → = ∅)
30 fvprc 6538 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
313, 30syl5eq 2845 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
32 mpoeq12 7092 . . . . 5 ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))))
3331, 31, 32syl2anc 584 . . . 4 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))))
34 mpo0 7102 . . . 4 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + (𝐼𝑦))) = ∅
3533, 34syl6eq 2849 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
3629, 35eqtr4d 2836 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
3727, 36pm2.61i 183 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1525  wcel 2083  Vcvv 3440  cun 3863  c0 4217  {csn 4478  cop 4484  ran crn 5451  cfv 6232  (class class class)co 7023  cmpo 7025  Basecbs 16316  +gcplusg 16398  invgcminusg 17866  -gcsg 17867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-sbg 17870
This theorem is referenced by:  grpsubval  17910  grpsubf  17939  grpsubpropd  17965  grpsubpropd2  17966  tgpsubcn  22386  tngtopn  22946
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