Theorem grpoinv 30728

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 30726	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈))
5	1, 2	grpoinveu 30722	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
6		riotacl2 7369	. . . . . 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 2862	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9		simpl 486	. . . . . . . . 9 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
10	9	rgenw 3080	. . . . . . . 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 30718	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
13	12	simprd 499	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
14	11, 13, 5	3jca 1141	. . . . . 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 4283	. . . . . 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 589	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
17	16	rabbidva 3420	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
18	8, 17	eleqtrd 2864	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19		oveq1 7403	. . . . . 6 ⊢ (𝑦 = (𝑁‘𝐴) → (𝑦𝐺𝐴) = ((𝑁‘𝐴)𝐺𝐴))
20	19	eqeq1d 2764	. . . . 5 ⊢ (𝑦 = (𝑁‘𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁‘𝐴)𝐺𝐴) = 𝑈))
21		oveq2 7404	. . . . . 6 ⊢ (𝑦 = (𝑁‘𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁‘𝐴)))
22	21	eqeq1d 2764	. . . . 5 ⊢ (𝑦 = (𝑁‘𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁‘𝐴)) = 𝑈))
23	20, 22	anbi12d 641	. . . 4 ⊢ (𝑦 = (𝑁‘𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
24	23	elrab 3650	. . 3 ⊢ ((𝑁‘𝐴) ∈ {𝑦 ∈ 𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁‘𝐴) ∈ 𝑋 ∧ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
25	18, 24	sylib 220	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ∈ 𝑋 ∧ (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)))
26	25	simprd 499	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  ∃!wreu 3365  {crab 3414  ran crn 5648  ‘cfv 6521  ℩crio 7352  (class class class)co 7396  GrpOpcgr 30692  GIdcgi 30693  invcgn 30694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-grpo 30696  df-gid 30697  df-ginv 30698

This theorem is referenced by:  grpolinv  30729  grporinv  30730