Theorem grpoinvid2 30824

Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (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
grpoinvid2	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Proof of Theorem grpoinvid2

Step	Hyp	Ref	Expression
1		oveq1 7420	. . . 4 ⊢ ((𝑁‘𝐴) = 𝐵 → ((𝑁‘𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
2	1	adantl 486	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → ((𝑁‘𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
3		grpinv.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4		grpinv.2	. . . . . 6 ⊢ 𝑈 = (GId‘𝐺)
5		grpinv.3	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
6	3, 4, 5	grpolinv 30821	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
7	6	3adant3 1148	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
8	7	adantr 485	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
9	2, 8	eqtr3d 2806	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
10	3, 5	grpoinvcl 30819	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
11	3, 4	grpolid 30811	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
12	10, 11	syldan 602	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
13	12	3adant3 1148	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
14	13	eqcomd 2775	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = (𝑈𝐺(𝑁‘𝐴)))
15	14	adantr 485	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁‘𝐴) = (𝑈𝐺(𝑁‘𝐴)))
16		oveq1 7420	. . . 4 ⊢ ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝑈𝐺(𝑁‘𝐴)))
17	16	adantl 486	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝑈𝐺(𝑁‘𝐴)))
18		simprr 784	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
19		simprl 782	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
20	10	adantrr 729	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘𝐴) ∈ 𝑋)
21	18, 19, 20	3jca 1144	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
22	3	grpoass 30798	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
23	21, 22	syldan 602	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
24	23	3impb 1130	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
25	3, 4, 5	grporinv 30822	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
26	25	oveq2d 7429	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
27	26	3adant3 1148	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
28	3, 4	grporid 30812	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
29	28	3adant2 1147	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
30	24, 27, 29	3eqtrd 2808	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = 𝐵)
31	30	adantr 485	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = 𝐵)
32	15, 17, 31	3eqtr2d 2810	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁‘𝐴) = 𝐵)
33	9, 32	impbida 812	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5665  ‘cfv 6539  (class class class)co 7413  GrpOpcgr 30784  GIdcgi 30785  invcgn 30786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-grpo 30788  df-gid 30789  df-ginv 30790

This theorem is referenced by:  rngonegmn1r  38518