Theorem grpinvval 12921

Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

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
grpinvval	⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦,𝑋

Allowed substitution hints:   + (𝑦)   𝑁(𝑦)   0 (𝑦)

Proof of Theorem grpinvval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2	1	basmex 12523	. . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ V)
3		grpinvval.p	. . . . 5 ⊢ + = (+g‘𝐺)
4		grpinvval.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
5		grpinvval.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
6	1, 3, 4, 5	grpinvfvalg 12920	. . . 4 ⊢ (𝐺 ∈ V → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
7	2, 6	syl 14	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )))
8	7	fveq1d 5519	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋))
9		eqid 2177	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
10		oveq2 5885	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
11	10	eqeq1d 2186	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
12	11	riotabidv 5835	. . 3 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
13		id 19	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵)
14		basfn 12522	. . . . . 6 ⊢ Base Fn V
15		funfvex 5534	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
16	15	funfni 5318	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
17	14, 2, 16	sylancr 414	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (Base‘𝐺) ∈ V)
18	1, 17	eqeltrid 2264	. . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐵 ∈ V)
19		riotaexg 5837	. . . 4 ⊢ (𝐵 ∈ V → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V)
20	18, 19	syl 14	. . 3 ⊢ (𝑋 ∈ 𝐵 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V)
21	9, 12, 13, 20	fvmptd3 5611	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
22	8, 21	eqtrd 2210	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ↦ cmpt 4066   Fn wfn 5213  ‘cfv 5218  ℩crio 5832  (class class class)co 5877  Basecbs 12464  +_gcplusg 12538  0_gc0g 12710  inv_gcminusg 12883

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-minusg 12886

This theorem is referenced by:  grplinv  12927  isgrpinv  12931