Theorem grpoinvop 30557

Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpasscan1.1	⊢ 𝑋 = ran 𝐺
grpasscan1.2	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpoinvop	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)))

Proof of Theorem grpoinvop

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simp2 1137	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1138	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
4		grpasscan1.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5		grpasscan1.2	. . . . . . 7 ⊢ 𝑁 = (inv‘𝐺)
6	4, 5	grpoinvcl 30548	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1131	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	4, 5	grpoinvcl 30548	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
9	8	3adant3 1132	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
10	4	grpocl 30524	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
11	1, 7, 9, 10	syl3anc 1373	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
12	4	grpoass 30527	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
13	1, 2, 3, 11, 12	syl13anc 1374	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
14		eqid 2734	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
15	4, 14, 5	grporinv 30551	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
16	15	3adant2 1131	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
17	16	oveq1d 7371	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = ((GId‘𝐺)𝐺(𝑁‘𝐴)))
18	4	grpoass 30527	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋)) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
19	1, 3, 7, 9, 18	syl13anc 1374	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
20	4, 14	grpolid 30540	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
21	8, 20	syldan 591	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
22	21	3adant3 1132	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
23	17, 19, 22	3eqtr3d 2777	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝑁‘𝐴))
24	23	oveq2d 7372	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))) = (𝐴𝐺(𝑁‘𝐴)))
25	4, 14, 5	grporinv 30551	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
26	25	3adant3 1132	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
27	13, 24, 26	3eqtrd 2773	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺))
28	4	grpocl 30524	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
29	4, 14, 5	grpoinvid1 30552	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺)))
30	1, 28, 11, 29	syl3anc 1373	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ↔ ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺)))
31	27, 30	mpbird 257	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ran crn 5623  ‘cfv 6490  (class class class)co 7356  GrpOpcgr 30513  GIdcgi 30514  invcgn 30515

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-grpo 30517  df-gid 30518  df-ginv 30519

This theorem is referenced by:  grpoinvdiv  30561