Theorem ablonnncan 26794

Description: Cancellation law for group division. (nnncan 10195 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
abldiv.1	⊢ 𝑋 = ran 𝐺
abldiv.3	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
ablonnncan	⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))

Proof of Theorem ablonnncan

Step	Hyp	Ref	Expression
1		simpr1 1060	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		ablogrpo 26785	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
3		abldiv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		abldiv.3	. . . . . . 7 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	3, 4	grpodivcl 26777	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
6	2, 5	syl3an1 1351	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
7	6	3adant3r1 1266	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) ∈ 𝑋)
8		simpr3 1062	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
9	1, 7, 8	3jca 1235	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
10	3, 4	ablodivdiv4 26792	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
11	9, 10	syldan 486	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)))
12	3, 4	grponpcan 26781	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
13	2, 12	syl3an1 1351	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
14	13	3adant3r1 1266	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵𝐷𝐶)𝐺𝐶) = 𝐵)
15	14	oveq2d 6565	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷((𝐵𝐷𝐶)𝐺𝐶)) = (𝐴𝐷𝐵))
16	11, 15	eqtrd 2644	1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727   /_𝑔 cgs 26730  AbelOpcablo 26782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783

This theorem is referenced by: (None)