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Theorem grpinvfvalg 12958
Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (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
grpinvfvalg (𝐺𝑉𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, 0   𝑥, +
Allowed substitution hints:   + (𝑦)   𝑁(𝑥,𝑦)   𝑉(𝑥,𝑦)   0 (𝑦)

Proof of Theorem grpinvfvalg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2 𝑁 = (invg𝐺)
2 df-minusg 12921 . . 3 invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))))
3 fveq2 5530 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 grpinvval.b . . . . 5 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2240 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6 fveq2 5530 . . . . . . . 8 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
7 grpinvval.p . . . . . . . 8 + = (+g𝐺)
86, 7eqtr4di 2240 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = + )
98oveqd 5908 . . . . . 6 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
10 fveq2 5530 . . . . . . 7 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
11 grpinvval.o . . . . . . 7 0 = (0g𝐺)
1210, 11eqtr4di 2240 . . . . . 6 (𝑔 = 𝐺 → (0g𝑔) = 0 )
139, 12eqeq12d 2204 . . . . 5 (𝑔 = 𝐺 → ((𝑦(+g𝑔)𝑥) = (0g𝑔) ↔ (𝑦 + 𝑥) = 0 ))
145, 13riotaeqbidv 5850 . . . 4 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔)) = (𝑦𝐵 (𝑦 + 𝑥) = 0 ))
155, 14mpteq12dv 4100 . . 3 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔)(𝑦(+g𝑔)𝑥) = (0g𝑔))) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
16 elex 2763 . . 3 (𝐺𝑉𝐺 ∈ V)
17 basfn 12544 . . . . . 6 Base Fn V
18 funfvex 5547 . . . . . . 7 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
1918funfni 5331 . . . . . 6 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
2017, 16, 19sylancr 414 . . . . 5 (𝐺𝑉 → (Base‘𝐺) ∈ V)
214, 20eqeltrid 2276 . . . 4 (𝐺𝑉𝐵 ∈ V)
2221mptexd 5759 . . 3 (𝐺𝑉 → (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )) ∈ V)
232, 15, 16, 22fvmptd3 5625 . 2 (𝐺𝑉 → (invg𝐺) = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
241, 23eqtrid 2234 1 (𝐺𝑉𝑁 = (𝑥𝐵 ↦ (𝑦𝐵 (𝑦 + 𝑥) = 0 )))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  cmpt 4079   Fn wfn 5226  cfv 5231  crio 5846  (class class class)co 5891  Basecbs 12486  +gcplusg 12561  0gc0g 12733  invgcminusg 12918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-ndx 12489  df-slot 12490  df-base 12492  df-minusg 12921
This theorem is referenced by:  grpinvval  12959  grpinvfng  12960  grpsubval  12962  grpinvf  12963  grpinvpropdg  12991  opprnegg  13400
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