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Theorem grpinvfval 18794
Description: The inverse function of a group. For a shorter proof using ax-rep 5243, see grpinvfvalALT 18795. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5243. (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 6843 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpinvval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6843 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpinvval.p . . . . . . . . 9 + = (+g𝐺)
75, 6eqtr4di 2791 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = + )
87oveqd 7375 . . . . . . 7 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
9 fveq2 6843 . . . . . . . 8 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
10 grpinvval.o . . . . . . . 8 0 = (0g𝐺)
119, 10eqtr4di 2791 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = 0 )
128, 11eqeq12d 2749 . . . . . 6 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
134, 12riotaeqbidv 7317 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
144, 13mpteq12dv 5197 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
15 df-minusg 18757 . . . 4 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
163fvexi 6857 . . . . 5 𝐵 ∈ V
17 p0ex 5340 . . . . . 6 {∅} ∈ V
1817, 16unex 7681 . . . . 5 ({∅} ∪ 𝐵) ∈ V
19 ssun2 4134 . . . . . . . 8 𝐵 ⊆ ({∅} ∪ 𝐵)
20 riotacl 7332 . . . . . . . 8 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
2119, 20sselid 3943 . . . . . . 7 (∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22 ssun1 4133 . . . . . . . 8 {∅} ⊆ ({∅} ∪ 𝐵)
23 riotaund 7354 . . . . . . . . 9 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24 riotaex 7318 . . . . . . . . . 10 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
2524elsn 4602 . . . . . . . . 9 ((𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (𝑦𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
2623, 25sylibr 233 . . . . . . . 8 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
2722, 26sselid 3943 . . . . . . 7 (¬ ∃!𝑦𝐵 (𝑦 + 𝑥) = 0 → (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
2821, 27pm2.61i 182 . . . . . 6 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
2928rgenw 3065 . . . . 5 𝑥𝐵 (𝑦𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
3016, 18, 29mptexw 7886 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
3114, 15, 30fvmpt 6949 . . 3 (𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
32 fvprc 6835 . . . . 5 𝐺 ∈ V → (invg𝐺) = ∅)
33 mpt0 6644 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = ∅
3432, 33eqtr4di 2791 . . . 4 𝐺 ∈ V → (invg𝐺) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
35 fvprc 6835 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
363, 35eqtrid 2785 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
3736mpteq1d 5201 . . . 4 𝐺 ∈ V → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3834, 37eqtr4d 2776 . . 3 𝐺 ∈ V → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
3931, 38pm2.61i 182 . 2 (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
401, 39eqtri 2761 1 𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  ∃!wreu 3350  Vcvv 3444  cun 3909  c0 4283  {csn 4587  cmpt 5189  cfv 6497  crio 7313  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  invgcminusg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-minusg 18757
This theorem is referenced by:  grpinvval  18796  grpinvfn  18797  grpinvf  18802  grpinvpropd  18827  opprneg  20069
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