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Theorem grpsubfval 18227
 Description: Group subtraction (division) operation. For a shorter proof using ax-rep 5160, see grpsubfvalALT 18228. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5160. (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 6663 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2811 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6663 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2811 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2759 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6663 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10eqtr4di 2811 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6665 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 7177 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpoeq123dv 7229 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 18187 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
163fvexi 6677 . . . . 5 𝐵 ∈ V
176fvexi 6677 . . . . . . 7 + ∈ V
1817rnex 7628 . . . . . 6 ran + ∈ V
19 p0ex 5257 . . . . . 6 {∅} ∈ V
2018, 19unex 7473 . . . . 5 (ran + ∪ {∅}) ∈ V
21 df-ov 7159 . . . . . . 7 (𝑥 + (𝐼𝑦)) = ( + ‘⟨𝑥, (𝐼𝑦)⟩)
22 fvrn0 6691 . . . . . . 7 ( + ‘⟨𝑥, (𝐼𝑦)⟩) ∈ (ran + ∪ {∅})
2321, 22eqeltri 2848 . . . . . 6 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2423rgen2w 3083 . . . . 5 𝑥𝐵𝑦𝐵 (𝑥 + (𝐼𝑦)) ∈ (ran + ∪ {∅})
2516, 16, 20, 24mpoexw 7787 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
2614, 15, 25fvmpt 6764 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
271, 26syl5eq 2805 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
28 fvprc 6655 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
291, 28syl5eq 2805 . . 3 𝐺 ∈ V → = ∅)
30 fvprc 6655 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
313, 30syl5eq 2805 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3231olcd 871 . . . 4 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
33 0mpo0 7237 . . . 4 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
3432, 33syl 17 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
3529, 34eqtr4d 2796 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
3627, 35pm2.61i 185 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∪ cun 3858  ∅c0 4227  {csn 4525  ⟨cop 4531  ran crn 5529  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16554  +gcplusg 16636  invgcminusg 18183  -gcsg 18184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-sbg 18187 This theorem is referenced by:  grpsubval  18229  grpsubf  18258  grpsubpropd  18284  grpsubpropd2  18285  tgpsubcn  22803  tngtopn  23365
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