Theorem grpo2inv 30619

Description: Double inverse law for groups. Lemma 2.2.1(c) 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
grpo2inv	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

Proof of Theorem grpo2inv

Step	Hyp	Ref	Expression
1		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
3	1, 2	grpoinvcl 30612	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
4		eqid 2737	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 4, 2	grporinv 30615	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
6	3, 5	syldan 592	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
7	1, 4, 2	grpolinv 30614	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	6, 7	eqtr4d 2775	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴))
9	1, 2	grpoinvcl 30612	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
10	3, 9	syldan 592	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
11		simpr 484	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
12	10, 11, 3	3jca 1129	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
13	1	grpolcan 30618	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋)) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
14	12, 13	syldan 592	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
15	8, 14	mpbid 232	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ran crn 5633  ‘cfv 6500  (class class class)co 7368  GrpOpcgr 30577  GIdcgi 30578  invcgn 30579

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-grpo 30581  df-gid 30582  df-ginv 30583

This theorem is referenced by:  grpoinvf  30620  grpodivinv  30624  grpoinvdiv  30625  nvnegneg  30737