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Theorem grpinvfval 18908
Description: The inverse function of a group. For a shorter proof using ax-rep 5278, see grpinvfvalALT 18909. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5278. (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 6885 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2784 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6885 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6eqtr4di 2784 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7422 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6885 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10eqtr4di 2784 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2742 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 7364 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 5232 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 18867 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
163fvexi 6899 . . . . 5 𝐵 ∈ V
17 p0ex 5375 . . . . . 6 {∅} ∈ V
1817, 16unex 7730 . . . . 5 ({∅} ∪ 𝐵) ∈ V
19 ssun2 4168 . . . . . . . 8 𝐵 ⊆ ({∅} ∪ 𝐵)
20 riotacl 7379 . . . . . . . 8 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
2119, 20sselid 3975 . . . . . . 7 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22 ssun1 4167 . . . . . . . 8 {∅} ⊆ ({∅} ∪ 𝐵)
23 riotaund 7401 . . . . . . . . 9 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24 riotaex 7365 . . . . . . . . . 10 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
2524elsn 4638 . . . . . . . . 9 ((𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
2623, 25sylibr 233 . . . . . . . 8 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
2722, 26sselid 3975 . . . . . . 7 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
2821, 27pm2.61i 182 . . . . . 6 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
2928rgenw 3059 . . . . 5 𝑥𝐵 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
3016, 18, 29mptexw 7938 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
3114, 15, 30fvmpt 6992 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
32 fvprc 6877 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
33 mpt0 6686 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
3432, 33eqtr4di 2784 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
35 fvprc 6877 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
363, 35eqtrid 2778 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3736mpteq1d 5236 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3834, 37eqtr4d 2769 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3931, 38pm2.61i 182 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
401, 39eqtri 2754 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  ∃!wreu 3368  Vcvv 3468  cun 3941  c0 4317  {csn 4623  cmpt 5224  cfv 6537  crio 7360  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  0gc0g 17394  invgcminusg 18864
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-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  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-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-riota 7361  df-ov 7408  df-minusg 18867
This theorem is referenced by:  grpinvval  18910  grpinvfn  18911  grpinvf  18916  grpinvpropd  18943  opprneg  20253
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