Theorem grpinvfvalg 13744

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 13706	. . 3 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
3		fveq2 5669	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpinvval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2283	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6		fveq2 5669	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
7		grpinvval.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8	6, 7	eqtr4di 2283	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
9	8	oveqd 6066	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
10		fveq2 5669	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
11		grpinvval.o	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	10, 11	eqtr4di 2283	. . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
13	9, 12	eqeq12d 2247	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
14	5, 13	riotaeqbidv 6005	. . . 4 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
15	5, 14	mpteq12dv 4191	. . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
16		elex 2824	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
17		basfn 13260	. . . . . 6 ⊢ Base Fn V
18		funfvex 5686	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
19	18	funfni 5457	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
20	17, 16, 19	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
21	4, 20	eqeltrid 2319	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
22	21	mptexd 5912	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V)
23	2, 15, 16, 22	fvmptd3 5770	. 2 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
24	1, 23	eqtrid 2277	1 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ↦ cmpt 4170   Fn wfn 5346  ‘cfv 5351  ℩crio 6001  (class class class)co 6049  Basecbs 13201  +_gcplusg 13279  0_gc0g 13458  inv_gcminusg 13703

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-minusg 13706

This theorem is referenced by:  grpinvval  13745  grpinvfng  13746  grpsubval  13748  grpinvf  13749  grpinvpropdg  13777  opprnegg  14216