Theorem grpoinvop 30604

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 1137	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simp2 1138	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1139	. . . 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 30595	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1132	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	4, 5	grpoinvcl 30595	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
9	8	3adant3 1133	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
10	4	grpocl 30571	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
11	1, 7, 9, 10	syl3anc 1374	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
12	4	grpoass 30574	. . . 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 1375	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
14		eqid 2736	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
15	4, 14, 5	grporinv 30598	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
16	15	3adant2 1132	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
17	16	oveq1d 7382	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = ((GId‘𝐺)𝐺(𝑁‘𝐴)))
18	4	grpoass 30574	. . . . . 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 1375	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
20	4, 14	grpolid 30587	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
21	8, 20	syldan 592	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
22	21	3adant3 1133	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
23	17, 19, 22	3eqtr3d 2779	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝑁‘𝐴))
24	23	oveq2d 7383	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))) = (𝐴𝐺(𝑁‘𝐴)))
25	4, 14, 5	grporinv 30598	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
26	25	3adant3 1133	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
27	13, 24, 26	3eqtrd 2775	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺))
28	4	grpocl 30571	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
29	4, 14, 5	grpoinvid1 30599	. . 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 1374	. 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 1087   = wceq 1542   ∈ wcel 2114  ran crn 5632  ‘cfv 6498  (class class class)co 7367  GrpOpcgr 30560  GIdcgi 30561  invcgn 30562

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-grpo 30564  df-gid 30565  df-ginv 30566

This theorem is referenced by:  grpoinvdiv  30608