Theorem ablonncan 28333

Description: Cancellation law for group division. (nncan 10915 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ablonncan	⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)

Proof of Theorem ablonncan

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
2	1	3anidm12 1415	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
3		abldiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		abldiv.3	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	3, 4	ablodivdiv 28330	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
6	2, 5	sylan2 594	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
7	6	3impb 1111	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8		ablogrpo 28324	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9		eqid 2821	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	3, 4, 9	grpodivid 28319	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
11	8, 10	sylan 582	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12	11	3adant3 1128	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
13	12	oveq1d 7171	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐴)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
14	3, 9	grpolid 28293	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
15	8, 14	sylan 582	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16	15	3adant2 1127	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
17	7, 13, 16	3eqtrd 2860	1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ran crn 5556  ‘cfv 6355  (class class class)co 7156  GrpOpcgr 28266  GIdcgi 28267   /_𝑔 cgs 28269  AbelOpcablo 28321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-grpo 28270  df-gid 28271  df-ginv 28272  df-gdiv 28273  df-ablo 28322

This theorem is referenced by:  ablonnncan1  28334