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Theorem grpsubfval 18913
Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5278, see grpsubfvalALT 18914. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5278. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)
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 6885 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2784 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6885 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2784 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2727 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6885 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10eqtr4di 2784 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6887 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 7426 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpoeq123dv 7480 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 18868 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
163fvexi 6899 . . . . 5 𝐵 ∈ V
176fvexi 6899 . . . . . . 7 + ∈ V
1817rnex 7900 . . . . . 6 ran + ∈ V
19 p0ex 5375 . . . . . 6 {∅} ∈ V
2018, 19unex 7730 . . . . 5 (ran + ∪ {∅}) ∈ V
21 df-ov 7408 . . . . . . 7 (𝑥 + (𝐼𝑦)) = ( + ‘⟨𝑥, (𝐼𝑦)⟩)
22 fvrn0 6915 . . . . . . 7 ( + ‘⟨𝑥, (𝐼𝑦)⟩) ∈ (ran + ∪ {∅})
2321, 22eqeltri 2823 . . . . . 6 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2423rgen2w 3060 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2516, 16, 20, 24mpoexw 8064 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
2614, 15, 25fvmpt 6992 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
271, 26eqtrid 2778 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
28 fvprc 6877 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
291, 28eqtrid 2778 . . 3 𝐺 ∈ V → = ∅)
30 fvprc 6877 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
313, 30eqtrid 2778 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3231olcd 871 . . . 4 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33 0mpo0 7488 . . . 4 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
3432, 33syl 17 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
3529, 34eqtr4d 2769 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
3627, 35pm2.61i 182 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1533  wcel 2098  Vcvv 3468  cun 3941  c0 4317  {csn 4623  cop 4629  ran crn 5670  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  +gcplusg 17206  invgcminusg 18864  -gcsg 18865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-sbg 18868
This theorem is referenced by:  grpsubval  18915  grpsubf  18947  grpsubpropd  18973  grpsubpropd2  18974  tgpsubcn  23949  tngtopn  24522
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