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Theorem grpinvfval 19009
Description: The inverse function of a group. For a shorter proof using ax-rep 5285, see grpinvfvalALT 19010. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5285. (Revised by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
grpinvval.b 𝐵 = (Base‘𝐺)
grpinvval.p + = (+g𝐺)
grpinvval.o 0 = (0g𝐺)
grpinvval.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvfval 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, 0   𝑥, +
Allowed substitution hints:   + (𝑦)   𝑁(𝑥,𝑦)   0 (𝑦)

Proof of Theorem grpinvfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2 𝑁 = (invg𝐺)
2 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6907 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6eqtr4di 2793 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7448 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6907 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10eqtr4di 2793 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2751 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 7391 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 5239 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 18968 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
163fvexi 6921 . . . . 5 𝐵 ∈ V
17 p0ex 5390 . . . . . 6 {∅} ∈ V
1817, 16unex 7763 . . . . 5 ({∅} ∪ 𝐵) ∈ V
19 ssun2 4189 . . . . . . . 8 𝐵 ⊆ ({∅} ∪ 𝐵)
20 riotacl 7405 . . . . . . . 8 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
2119, 20sselid 3993 . . . . . . 7 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22 ssun1 4188 . . . . . . . 8 {∅} ⊆ ({∅} ∪ 𝐵)
23 riotaund 7427 . . . . . . . . 9 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24 riotaex 7392 . . . . . . . . . 10 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
2524elsn 4646 . . . . . . . . 9 ((𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
2623, 25sylibr 234 . . . . . . . 8 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
2722, 26sselid 3993 . . . . . . 7 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
2821, 27pm2.61i 182 . . . . . 6 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
2928rgenw 3063 . . . . 5 𝑥𝐵 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
3016, 18, 29mptexw 7976 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
3114, 15, 30fvmpt 7016 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
32 fvprc 6899 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
33 mpt0 6711 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
3432, 33eqtr4di 2793 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
35 fvprc 6899 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
363, 35eqtrid 2787 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3736mpteq1d 5243 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3834, 37eqtr4d 2778 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3931, 38pm2.61i 182 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
401, 39eqtri 2763 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  ∃!wreu 3376  Vcvv 3478  cun 3961  c0 4339  {csn 4631  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  invgcminusg 18965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-minusg 18968
This theorem is referenced by:  grpinvval  19011  grpinvfn  19012  grpinvf  19017  grpinvpropd  19046  opprneg  20368
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