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Theorem grpinvfval 18155
 Description: The inverse function of a group. For a shorter proof using ax-rep 5158, see grpinvfvalALT 18156. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5158. (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 6655 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2851 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6655 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6eqtr4di 2851 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7162 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6655 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10eqtr4di 2851 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2814 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 7106 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 5119 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 18119 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
163fvexi 6669 . . . . 5 𝐵 ∈ V
17 p0ex 5254 . . . . . 6 {∅} ∈ V
1817, 16unex 7462 . . . . 5 ({∅} ∪ 𝐵) ∈ V
19 ssun2 4103 . . . . . . . 8 𝐵 ⊆ ({∅} ∪ 𝐵)
20 riotacl 7120 . . . . . . . 8 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
2119, 20sseldi 3915 . . . . . . 7 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22 ssun1 4102 . . . . . . . 8 {∅} ⊆ ({∅} ∪ 𝐵)
23 riotaund 7142 . . . . . . . . 9 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24 riotaex 7107 . . . . . . . . . 10 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
2524elsn 4543 . . . . . . . . 9 ((𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
2623, 25sylibr 237 . . . . . . . 8 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
2722, 26sseldi 3915 . . . . . . 7 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
2821, 27pm2.61i 185 . . . . . 6 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
2928rgenw 3118 . . . . 5 𝑥𝐵 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
3016, 18, 29mptexw 7649 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
3114, 15, 30fvmpt 6755 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
32 fvprc 6647 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
33 mpt0 6470 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
3432, 33eqtr4di 2851 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
35 fvprc 6647 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
363, 35syl5eq 2845 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3736mpteq1d 5123 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3834, 37eqtr4d 2836 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3931, 38pm2.61i 185 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
401, 39eqtri 2821 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  ∃!wreu 3108  Vcvv 3442   ∪ cun 3881  ∅c0 4246  {csn 4528   ↦ cmpt 5114  ‘cfv 6332  ℩crio 7102  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  0gc0g 16725  invgcminusg 18116 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 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-riota 7103  df-ov 7148  df-minusg 18119 This theorem is referenced by:  grpinvval  18157  grpinvfn  18158  grpinvf  18163  grpinvpropd  18187  opprneg  19402
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