Theorem grpoinvop 28316

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 1133	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simp2 1134	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1135	. . . 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 28307	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1128	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	4, 5	grpoinvcl 28307	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
9	8	3adant3 1129	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
10	4	grpocl 28283	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
11	1, 7, 9, 10	syl3anc 1368	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
12	4	grpoass 28286	. . . 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 1369	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
14		eqid 2798	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
15	4, 14, 5	grporinv 28310	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
16	15	3adant2 1128	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
17	16	oveq1d 7150	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = ((GId‘𝐺)𝐺(𝑁‘𝐴)))
18	4	grpoass 28286	. . . . . 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 1369	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
20	4, 14	grpolid 28299	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
21	8, 20	syldan 594	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
22	21	3adant3 1129	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
23	17, 19, 22	3eqtr3d 2841	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝑁‘𝐴))
24	23	oveq2d 7151	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))) = (𝐴𝐺(𝑁‘𝐴)))
25	4, 14, 5	grporinv 28310	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
26	25	3adant3 1129	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
27	13, 24, 26	3eqtrd 2837	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺))
28	4	grpocl 28283	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
29	4, 14, 5	grpoinvid1 28311	. . 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 1368	. 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 260	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5520  ‘cfv 6324  (class class class)co 7135  GrpOpcgr 28272  GIdcgi 28273  invcgn 28274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-grpo 28276  df-gid 28277  df-ginv 28278

This theorem is referenced by:  grpoinvdiv  28320