Theorem grpoinvdiv 30523

Description: Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdiv.1	⊢ 𝑋 = ran 𝐺
grpdiv.2	⊢ 𝑁 = (inv‘𝐺)
grpdiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
grpoinvdiv	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))

Proof of Theorem grpoinvdiv

Step	Hyp	Ref	Expression
1		grpdiv.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		grpdiv.2	. . . 4 ⊢ 𝑁 = (inv‘𝐺)
3		grpdiv.3	. . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	1, 2, 3	grpodivval 30521	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))
5	4	fveq2d 6885	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁‘𝐵))))
6	1, 2	grpoinvcl 30510	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1131	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	1, 2	grpoinvop 30519	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)))
9	7, 8	syld3an3 1411	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)))
10	1, 2	grpo2inv 30517	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
11	10	3adant2 1131	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
12	11	oveq1d 7425	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝑁‘𝐴)))
13	1, 2, 3	grpodivval 30521	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴)))
14	13	3com23 1126	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴)))
15	12, 14	eqtr4d 2774	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐷𝐴))
16	5, 9, 15	3eqtrd 2775	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5660  ‘cfv 6536  (class class class)co 7410  GrpOpcgr 30475  invcgn 30477   /_𝑔 cgs 30478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-grpo 30479  df-gid 30480  df-ginv 30481  df-gdiv 30482

This theorem is referenced by:  grpodivdiv  30526