Theorem grpinvfval 18937

Description: The inverse function of a group. For a shorter proof using ax-rep 5280, see grpinvfvalALT 18938. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) Remove dependency on ax-rep 5280. (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 6891	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		grpinvval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6891	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		grpinvval.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7432	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥))
9		fveq2 6891	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
10		grpinvval.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
11	9, 10	eqtr4di 2783	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
12	8, 11	eqeq12d 2741	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 ))
13	4, 12	riotaeqbidv 7374	. . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
14	4, 13	mpteq12dv 5234	. . . 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 18896	. . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))))
16	3	fvexi 6905	. . . . 5 ⊢ 𝐵 ∈ V
17		p0ex 5378	. . . . . 6 ⊢ {∅} ∈ V
18	17, 16	unex 7745	. . . . 5 ⊢ ({∅} ∪ 𝐵) ∈ V
19		ssun2 4167	. . . . . . . 8 ⊢ 𝐵 ⊆ ({∅} ∪ 𝐵)
20		riotacl 7389	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ 𝐵)
21	19, 20	sselid 3970	. . . . . . 7 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵))
22		ssun1 4166	. . . . . . . 8 ⊢ {∅} ⊆ ({∅} ∪ 𝐵)
23		riotaund 7411	. . . . . . . . 9 ⊢ (¬ ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = ∅)
24		riotaex 7375	. . . . . . . . . 10 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ V
25	24	elsn 4639	. . . . . . . . 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 3970	. . . . . . 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 3055	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) ∈ ({∅} ∪ 𝐵)
30	16, 18, 29	mptexw 7952	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V
31	14, 15, 30	fvmpt 6999	. . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
32		fvprc 6883	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅)
33		mpt0 6691	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅
34	32, 33	eqtr4di 2783	. . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
35		fvprc 6883	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
36	3, 35	eqtrid 2777	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
37	36	mpteq1d 5238	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
38	34, 37	eqtr4d 2768	. . 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 2753	1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  ∃!wreu 3362  Vcvv 3463   ∪ cun 3938  ∅c0 4318  {csn 4624   ↦ cmpt 5226  ‘cfv 6542  ℩crio 7370  (class class class)co 7415  Basecbs 17177  +_gcplusg 17230  0_gc0g 17418  inv_gcminusg 18893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7371  df-ov 7418  df-minusg 18896

This theorem is referenced by:  grpinvval  18939  grpinvfn  18940  grpinvf  18945  grpinvpropd  18973  opprneg  20292