Theorem grpo2inv 31013

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 31006	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
4		eqid 2760	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 4, 2	grporinv 31009	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
6	3, 5	syldan 603	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
7	1, 4, 2	grpolinv 31008	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	6, 7	eqtr4d 2798	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴))
9	1, 2	grpoinvcl 31006	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
10	3, 9	syldan 603	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
11		simpr 490	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
12	10, 11, 3	3jca 1146	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
13	1	grpolcan 31012	. . 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 603	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
15	8, 14	mpbid 235	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ran crn 5656  ‘cfv 6533  (class class class)co 7414  GrpOpcgr 30971  GIdcgi 30972  invcgn 30973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-grpo 30975  df-gid 30976  df-ginv 30977

This theorem is used by:  grpoinvf  31014  grpodivinv  31018  grpoinvdiv  31019  nvnegneg  31131