Theorem grpoinvid1 30624

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
grpoinvid1	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Proof of Theorem grpoinvid1

Step	Hyp	Ref	Expression
1		oveq2 7371	. . . 4 ⊢ ((𝑁‘𝐴) = 𝐵 → (𝐴𝐺(𝑁‘𝐴)) = (𝐴𝐺𝐵))
2	1	adantl 482	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺(𝑁‘𝐴)) = (𝐴𝐺𝐵))
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	grporinv 30623	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
7	6	3adant3 1138	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
8	7	adantr 481	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
9	2, 8	eqtr3d 2777	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10		oveq2 7371	. . . 4 ⊢ ((𝐴𝐺𝐵) = 𝑈 → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁‘𝐴)𝐺𝑈))
11	10	adantl 482	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁‘𝐴)𝐺𝑈))
12	3, 4, 5	grpolinv 30622	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
13	12	oveq1d 7378	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14	13	3adant3 1138	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
15	3, 5	grpoinvcl 30620	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
16	15	adantrr 723	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘𝐴) ∈ 𝑋)
17		simprl 776	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
18		simprr 778	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
19	16, 17, 18	3jca 1134	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
20	3	grpoass 30599	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑁‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
21	19, 20	syldan 597	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
22	21	3impb 1120	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
23	14, 22	eqtr3d 2777	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
24	3, 4	grpolid 30612	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
25	24	3adant2 1137	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
26	23, 25	eqtr3d 2777	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
27	26	adantr 481	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
28	3, 4	grporid 30613	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
29	15, 28	syldan 597	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
30	29	3adant3 1138	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
31	30	adantr 481	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
32	11, 27, 31	3eqtr3rd 2784	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁‘𝐴) = 𝐵)
33	9, 32	impbida 806	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ran crn 5626  ‘cfv 6492  (class class class)co 7363  GrpOpcgr 30585  GIdcgi 30586  invcgn 30587

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-grpo 30589  df-gid 30590  df-ginv 30591

This theorem is referenced by:  grpoinvop  30629  rngonegmn1l  38315