Theorem grpinvfval 18533

Description: The inverse function of a group. For a shorter proof using ax-rep 5205, see grpinvfvalALT 18534. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5205. (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
grpinvfval	⊢ 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)   0 (𝑦)

Proof of Theorem grpinvfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvval.n	. 2 ⊢ 𝑁 = (invg‘𝐺)
2		fveq2 6756	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpinvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2797	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6756	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpinvval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7272	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
9		fveq2 6756	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		grpinvval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2797	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2754	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
13	4, 12	riotaeqbidv 7215	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
14	4, 13	mpteq12dv 5161	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
15		df-minusg 18496	. . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
16	3	fvexi 6770	. . . . 5 ⊢ 𝐵 ∈ V
17		p0ex 5302	. . . . . 6 ⊢ {∅} ∈ V
18	17, 16	unex 7574	. . . . 5 ⊢ ({∅} ∪ 𝐵) ∈ V
19		ssun2 4103	. . . . . . . 8 ⊢ 𝐵 ⊆ ({∅} ∪ 𝐵)
20		riotacl 7230	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
21	19, 20	sselid 3915	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22		ssun1 4102	. . . . . . . 8 ⊢ {∅} ⊆ ({∅} ∪ 𝐵)
23		riotaund 7252	. . . . . . . . 9 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24		riotaex 7216	. . . . . . . . . 10 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
25	24	elsn 4573	. . . . . . . . 9 ⊢ ((℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
26	23, 25	sylibr 233	. . . . . . . 8 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
27	22, 26	sselid 3915	. . . . . . 7 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
28	21, 27	pm2.61i 182	. . . . . 6 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
29	28	rgenw 3075	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
30	16, 18, 29	mptexw 7769	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
31	14, 15, 30	fvmpt 6857	. . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
32		fvprc 6748	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅)
33		mpt0 6559	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅
34	32, 33	eqtr4di 2797	. . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
35		fvprc 6748	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
36	3, 35	eqtrid 2790	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
37	36	mpteq1d 5165	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
38	34, 37	eqtr4d 2781	. . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
39	31, 38	pm2.61i 182	. 2 ⊢ (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
40	1, 39	eqtri 2766	1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  ∃!wreu 3065  Vcvv 3422   ∪ cun 3881  ∅c0 4253  {csn 4558   ↦ cmpt 5153  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  inv_gcminusg 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-minusg 18496

This theorem is referenced by:  grpinvval  18535  grpinvfn  18536  grpinvf  18541  grpinvpropd  18565  opprneg  19792