Theorem grpoinvdiv 30566

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 30564	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵)))
5	4	fveq2d 6911	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁‘𝐵))))
6	1, 2	grpoinvcl 30553	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
7	6	3adant2 1130	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
8	1, 2	grpoinvop 30562	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)))
9	7, 8	syld3an3 1408	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)))
10	1, 2	grpo2inv 30560	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
11	10	3adant2 1130	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
12	11	oveq1d 7446	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝑁‘𝐴)))
13	1, 2, 3	grpodivval 30564	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴)))
14	13	3com23 1125	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴)))
15	12, 14	eqtr4d 2778	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐷𝐴))
16	5, 9, 15	3eqtrd 2779	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ran crn 5690  ‘cfv 6563  (class class class)co 7431  GrpOpcgr 30518  invcgn 30520   /_𝑔 cgs 30521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ginv 30524  df-gdiv 30525

This theorem is referenced by:  grpodivdiv  30569