Theorem grpinvval 13536

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 13052	. . . 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 13535	. . . 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 5602	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋))
9		eqid 2207	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
10		oveq2 5977	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
11	10	eqeq1d 2216	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
12	11	riotabidv 5926	. . 3 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
13		id 19	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵)
14		basfn 13051	. . . . . 6 ⊢ Base Fn V
15		funfvex 5617	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
16	15	funfni 5396	. . . . . 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 2294	. . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐵 ∈ V)
19		riotaexg 5928	. . . 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 5698	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
22	8, 21	eqtrd 2240	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  Vcvv 2777   ↦ cmpt 4122   Fn wfn 5286  ‘cfv 5291  ℩crio 5923  (class class class)co 5969  Basecbs 12993  +_gcplusg 13070  0_gc0g 13249  inv_gcminusg 13494

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-cnex 8053  ax-resscn 8054  ax-1re 8056  ax-addrcl 8059

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-riota 5924  df-ov 5972  df-inn 9074  df-ndx 12996  df-slot 12997  df-base 12999  df-minusg 13497

This theorem is referenced by:  grplinv  13543  isgrpinv  13547