Theorem grpoinv 30614

Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinv.1	⊢ 𝑋 = ran 𝐺
grpinv.2	⊢ 𝑈 = (GId‘𝐺)
grpinv.3	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpoinv	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈))

Proof of Theorem grpoinv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinv.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
2		grpinv.2	. . . . . 6 ⊢ 𝑈 = (GId‘𝐺)
3		grpinv.3	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
4	1, 2, 3	grpoinvval 30612	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
5	1, 2	grpoinveu 30608	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
6		riotacl2 7329	. . . . . 6 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
7	5, 6	syl 17	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
8	4, 7	eqeltrd 2839	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9		simpl 483	. . . . . . . . 9 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
10	9	rgenw 3057	. . . . . . . 8 ⊢ ∀𝑦 ∈ 𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
11	10	a1i 11	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
12	1, 2	grpoidinv2 30604	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
13	12	simprd 496	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
14	11, 13, 5	3jca 1134	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
15		reupick2 4259	. . . . . 6 ⊢ (((∀𝑦 ∈ 𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
16	14, 15	sylan 586	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
17	16	rabbidva 3397	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
18	8, 17	eleqtrd 2841	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19		oveq1 7363	. . . . . 6 ⊢ (𝑦 = (𝑁‘𝐴) → (𝑦𝐺𝐴) = ((𝑁‘𝐴)𝐺𝐴))
20	19	eqeq1d 2741	. . . . 5 ⊢ (𝑦 = (𝑁‘𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁‘𝐴)𝐺𝐴) = 𝑈))
21		oveq2 7364	. . . . . 6 ⊢ (𝑦 = (𝑁‘𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁‘𝐴)))
22	21	eqeq1d 2741	. . . . 5 ⊢ (𝑦 = (𝑁‘𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁‘𝐴)) = 𝑈))
23	20, 22	anbi12d 638	. . . 4 ⊢ (𝑦 = (𝑁‘𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
24	23	elrab 3629	. . 3 ⊢ ((𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁‘𝐴) ∈ 𝑋 ∧ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
25	18, 24	sylib 219	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ∈ 𝑋 ∧ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
26	25	simprd 496	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  ∃!wreu 3342  {crab 3391  ran crn 5619  ‘cfv 6485  ℩crio 7312  (class class class)co 7356  GrpOpcgr 30578  GIdcgi 30579  invcgn 30580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-grpo 30582  df-gid 30583  df-ginv 30584

This theorem is referenced by:  grpolinv  30615  grporinv  30616