Theorem grpinvval 13850

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 13414	. . . 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 13849	. . . 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 5697	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋))
9		eqid 2238	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
10		oveq2 6093	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
11	10	eqeq1d 2247	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
12	11	riotabidv 6040	. . 3 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
13		id 19	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵)
14		basfn 13413	. . . . . 6 ⊢ Base Fn V
15		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
16	15	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
17	14, 2, 16	sylancr 418	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (Base‘𝐺) ∈ V)
18	1, 17	eqeltrid 2325	. . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐵 ∈ V)
19		riotaexg 6042	. . . 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 5799	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
22	8, 21	eqtrd 2271	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4192   Fn wfn 5372  ‘cfv 5377  ℩crio 6037  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  0_gc0g 13612  inv_gcminusg 13808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9306  df-ndx 13357  df-slot 13358  df-base 13360  df-minusg 13811

This theorem is used by:  grplinv  13857  isgrpinv  13861