Theorem ablo4pnp 37864

Description: A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)

Hypotheses

Ref	Expression
abl4pnp.1	⊢ 𝑋 = ran 𝐺
abl4pnp.2	⊢ 𝐷 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
ablo4pnp	⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Proof of Theorem ablo4pnp

Step	Hyp	Ref	Expression
1		df-3an 1088	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋))
2		abl4pnp.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
3		abl4pnp.2	. . . . . 6 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
4	2, 3	ablomuldiv 30496	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
5	1, 4	sylan2br 595	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
6	5	adantrrr 725	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
7	6	oveq1d 7364	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8		ablogrpo 30491	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9	2	grpocl 30444	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
10	9	3expib 1122	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
11	8, 10	syl 17	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
12	11	anim1d 611	. . . . 5 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
13		3anass 1094	. . . . 5 ⊢ (((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
14	12, 13	imbitrrdi 252	. . . 4 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
15	14	imp 406	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
16	2, 3	ablodivdiv4 30498	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
17	15, 16	syldan 591	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
18	2, 3	grpodivcl 30483	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19	18	3expib 1122	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
20	19	anim1d 611	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
21		an4 656	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
22		3anass 1094	. . . . . 6 ⊢ (((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
23	20, 21, 22	3imtr4g 296	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
24	23	imp 406	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
25	2, 3	grpomuldivass 30485	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
26	24, 25	syldan 591	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
27	8, 26	sylan 580	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
28	7, 17, 27	3eqtr3d 2772	1 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5620  ‘cfv 6482  (class class class)co 7349  GrpOpcgr 30433   /_𝑔 cgs 30436  AbelOpcablo 30488

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-grpo 30437  df-gid 30438  df-ginv 30439  df-gdiv 30440  df-ablo 30489

This theorem is referenced by: (None)