Theorem grpoinvop 30622

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 1142	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simp2 1143	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1144	. . . 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 30613	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1137	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	4, 5	grpoinvcl 30613	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
9	8	3adant3 1138	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
10	4	grpocl 30589	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
11	1, 7, 9, 10	syl3anc 1379	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
12	4	grpoass 30592	. . . 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 1380	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
14		eqid 2739	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
15	4, 14, 5	grporinv 30616	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
16	15	3adant2 1137	. . . . . 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 30592	. . . . . 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 1380	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
20	4, 14	grpolid 30605	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
21	8, 20	syldan 597	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
22	21	3adant3 1138	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
23	17, 19, 22	3eqtr3d 2782	. . . 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 30616	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
26	25	3adant3 1138	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
27	13, 24, 26	3eqtrd 2778	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺))
28	4	grpocl 30589	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
29	4, 14, 5	grpoinvid1 30617	. . 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 1379	. 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 258	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ran crn 5619  ‘cfv 6485  (class class class)co 7356  GrpOpcgr 30578  GIdcgi 30579  invcgn 30580

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 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

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 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-grpo 30582  df-gid 30583  df-ginv 30584

This theorem is referenced by:  grpoinvdiv  30626