Theorem grpoinvid2 28308

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 7165	. . . 4 ⊢ ((𝑁‘𝐴) = 𝐵 → ((𝑁‘𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
2	1	adantl 484	. . 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 28305	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
7	6	3adant3 1128	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
8	7	adantr 483	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
9	2, 8	eqtr3d 2860	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
10	3, 5	grpoinvcl 28303	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
11	3, 4	grpolid 28295	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
12	10, 11	syldan 593	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
13	12	3adant3 1128	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
14	13	eqcomd 2829	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = (𝑈𝐺(𝑁‘𝐴)))
15	14	adantr 483	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁‘𝐴) = (𝑈𝐺(𝑁‘𝐴)))
16		oveq1 7165	. . . 4 ⊢ ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝑈𝐺(𝑁‘𝐴)))
17	16	adantl 484	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝑈𝐺(𝑁‘𝐴)))
18		simprr 771	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
19		simprl 769	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
20	10	adantrr 715	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘𝐴) ∈ 𝑋)
21	18, 19, 20	3jca 1124	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
22	3	grpoass 28282	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
23	21, 22	syldan 593	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
24	23	3impb 1111	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))))
25	3, 4, 5	grporinv 28306	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
26	25	oveq2d 7174	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
27	26	3adant3 1128	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝐴𝐺(𝑁‘𝐴))) = (𝐵𝐺𝑈))
28	3, 4	grporid 28296	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
29	28	3adant2 1127	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺𝑈) = 𝐵)
30	24, 27, 29	3eqtrd 2862	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = 𝐵)
31	30	adantr 483	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁‘𝐴)) = 𝐵)
32	15, 17, 31	3eqtr2d 2864	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁‘𝐴) = 𝐵)
33	9, 32	impbida 799	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ran crn 5558  ‘cfv 6357  (class class class)co 7158  GrpOpcgr 28268  GIdcgi 28269  invcgn 28270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-grpo 28272  df-gid 28273  df-ginv 28274

This theorem is referenced by:  rngonegmn1r  35222