Theorem grpoinvop 27939

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 1170	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simp2 1171	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simp3 1172	. . . 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 27930	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1165	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	4, 5	grpoinvcl 27930	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
9	8	3adant3 1166	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
10	4	grpocl 27906	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐵) ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
11	1, 7, 9, 10	syl3anc 1494	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐵)𝐺(𝑁‘𝐴)) ∈ 𝑋)
12	4	grpoass 27909	. . . 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 1495	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))))
14		eqid 2825	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
15	4, 14, 5	grporinv 27933	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
16	15	3adant2 1165	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(𝑁‘𝐵)) = (GId‘𝐺))
17	16	oveq1d 6925	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = ((GId‘𝐺)𝐺(𝑁‘𝐴)))
18	4	grpoass 27909	. . . . . 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 1495	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐵𝐺(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))))
20	4, 14	grpolid 27922	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
21	8, 20	syldan 585	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
22	21	3adant3 1166	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺(𝑁‘𝐴)) = (𝑁‘𝐴))
23	17, 19, 22	3eqtr3d 2869	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (𝑁‘𝐴))
24	23	oveq2d 6926	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝐵𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴)))) = (𝐴𝐺(𝑁‘𝐴)))
25	4, 14, 5	grporinv 27933	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
26	25	3adant3 1166	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = (GId‘𝐺))
27	13, 24, 26	3eqtrd 2865	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺((𝑁‘𝐵)𝐺(𝑁‘𝐴))) = (GId‘𝐺))
28	4	grpocl 27906	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
29	4, 14, 5	grpoinvid1 27934	. . 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 1494	. 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 249	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁‘𝐵)𝐺(𝑁‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ran crn 5347  ‘cfv 6127  (class class class)co 6910  GrpOpcgr 27895  GIdcgi 27896  invcgn 27897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-grpo 27899  df-gid 27900  df-ginv 27901

This theorem is referenced by:  grpoinvdiv  27943