Theorem grpoinvid2 30890

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 7417	. . . 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 30887	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
7	6	3adant3 1150	. . . 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 2800	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
10	3, 5	grpoinvcl 30885	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
11	3, 4	grpolid 30877	. . . . . . 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 1150	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
14	13	eqcomd 2769	. . . 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 7417	. . . 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 1146	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
22	3	grpoass 30864	. . . . . . 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 1132	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
25	3, 4, 5	grporinv 30888	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
26	25	oveq2d 7426	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
27	26	3adant3 1150	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
28	3, 4	grporid 30878	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
29	28	3adant2 1149	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
30	24, 27, 29	3eqtrd 2802	. . . 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 2804	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ran crn 5662  ‘cfv 6536  (class class class)co 7410  GrpOpcgr 30850  GIdcgi 30851  invcgn 30852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-grpo 30854  df-gid 30855  df-ginv 30856

This theorem is used by:  rngonegmn1r  38621