Theorem grpinvfval 19046

Description: The inverse function of a group. For a shorter proof using ax-rep 5239, see grpinvfvalALT 19047. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5239. (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 6883	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpinvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6883	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpinvval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7429	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
9		fveq2 6883	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		grpinvval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2779	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
13	4, 12	riotaeqbidv 7372	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
14	4, 13	mpteq12dv 5199	. . . 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 19005	. . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
16	3	fvexi 6897	. . . . 5 ⊢ 𝐵 ∈ V
17		p0ex 5357	. . . . . 6 ⊢ {∅} ∈ V
18	17, 16	unex 7744	. . . . 5 ⊢ ({∅} ∪ 𝐵) ∈ V
19		ssun2 4133	. . . . . . . 8 ⊢ 𝐵 ⊆ ({∅} ∪ 𝐵)
20		riotacl 7386	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
21	19, 20	sselid 3936	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22		ssun1 4132	. . . . . . . 8 ⊢ {∅} ⊆ ({∅} ∪ 𝐵)
23		riotaund 7408	. . . . . . . . 9 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24		riotaex 7373	. . . . . . . . . 10 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
25	24	elsn 4605	. . . . . . . . 9 ⊢ ((℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅} ↔ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
26	23, 25	sylibr 237	. . . . . . . 8 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ {∅})
27	22, 26	sselid 3936	. . . . . . 7 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
28	21, 27	pm2.61i 184	. . . . . 6 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
29	28	rgenw 3083	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
30	16, 18, 29	mptexw 7951	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
31	14, 15, 30	fvmpt 6991	. . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
32		fvprc 6875	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅)
33		mpt0 6679	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅
34	32, 33	eqtr4di 2816	. . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
35		fvprc 6875	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
36	3, 35	eqtrid 2810	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
37	36	mpteq1d 5202	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
38	34, 37	eqtr4d 2801	. . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
39	31, 38	pm2.61i 184	. 2 ⊢ (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
40	1, 39	eqtri 2786	1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  ∃!wreu 3367  Vcvv 3455   ∪ cun 3904  ∅c0 4287  {csn 4590   ↦ cmpt 5193  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493  inv_gcminusg 19002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7369  df-ov 7415  df-minusg 19005

This theorem is referenced by:  grpinvval  19048  grpinvfn  19049  grpinvf  19054  grpinvpropd  19082  opprneg  20434