Theorem grpinvfvalg 12745

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 12712	. . 3 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
3		fveq2 5496	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpinvval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2221	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5496	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpinvval.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2221	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 5870	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
10		fveq2 5496	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
11		grpinvval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	10, 11	eqtr4di 2221	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
13	9, 12	eqeq12d 2185	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
14	5, 13	riotaeqbidv 5812	. . . 4 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
15	5, 14	mpteq12dv 4071	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
16		elex 2741	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
17		basfn 12473	. . . . . 6 ⊢ Base Fn V
18		funfvex 5513	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
19	18	funfni 5298	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
20	17, 16, 19	sylancr 412	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
21	4, 20	eqeltrid 2257	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
22	21	mptexd 5723	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V)
23	2, 15, 16, 22	fvmptd3 5589	. 2 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
24	1, 23	eqtrid 2215	1 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ↦ cmpt 4050   Fn wfn 5193  ‘cfv 5198  ℩crio 5808  (class class class)co 5853  Basecbs 12416  +_gcplusg 12480  0_gc0g 12596  inv_gcminusg 12709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-minusg 12712

This theorem is referenced by:  grpinvval  12746  grpinvfng  12747  grpsubval  12749  grpinvf  12750