Theorem grpinvfvalg 13830

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	⊢ + = (+g‘𝐺)
grpinvval.o	⊢ 0 = (0g‘𝐺)
grpinvval.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvfvalg	⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (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.

Step	Hyp	Ref	Expression
1		grpinvval.n	. 2 ⊢ 𝑁 = (invg‘𝐺)
2		df-minusg 13792	. . 3 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
3		fveq2 5693	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpinvval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2289	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5693	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpinvval.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2289	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 6096	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
10		fveq2 5693	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
11		grpinvval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	10, 11	eqtr4di 2289	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
13	9, 12	eqeq12d 2253	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
14	5, 13	riotaeqbidv 6035	. . . 4 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
15	5, 14	mpteq12dv 4211	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
16		elex 2833	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
17		basfn 13394	. . . . . 6 ⊢ Base Fn V
18		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
19	18	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
20	17, 16, 19	sylancr 418	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
21	4, 20	eqeltrid 2325	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
22	21	mptexd 5938	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V)
23	2, 15, 16, 22	fvmptd3 5796	. 2 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
24	1, 23	eqtrid 2283	1 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4190   Fn wfn 5370  ‘cfv 5375  ℩crio 6031  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  inv_gcminusg 13789

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-minusg 13792

This theorem is referenced by:  grpinvval  13831  grpinvfng  13832  grpsubval  13834  grpinvf  13835  grpinvpropdg  13863  opprnegg  14372