Theorem ablonncan 28451

Description: Cancellation law for group division. (nncan 10966 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 1416	. . . 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 28448	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
6	2, 5	sylan2 595	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
7	6	3impb 1112	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = ((𝐴𝐷𝐴)𝐺𝐵))
8		ablogrpo 28442	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9		eqid 2758	. . . . . 6 ⊢ (GId‘𝐺) = (GId‘𝐺)
10	3, 4, 9	grpodivid 28437	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
11	8, 10	sylan 583	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (GId‘𝐺))
12	11	3adant3 1129	. . 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 28411	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
15	8, 14	sylan 583	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
16	15	3adant2 1128	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
17	7, 13, 16	3eqtrd 2797	1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5529  ‘cfv 6340  (class class class)co 7156  GrpOpcgr 28384  GIdcgi 28385   /_𝑔 cgs 28387  AbelOpcablo 28439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-grpo 28388  df-gid 28389  df-ginv 28390  df-gdiv 28391  df-ablo 28440

This theorem is referenced by:  ablonnncan1  28452