Theorem ablo4pnp 38453

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 1103	. . . . 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 30845	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
5	1, 4	sylan2br 606	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
6	5	adantrrr 737	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
7	6	oveq1d 7426	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8		ablogrpo 30840	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9	2	grpocl 30793	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
10	9	3expib 1138	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
11	8, 10	syl 18	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
12	11	anim1d 622	. . . . 5 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
13		3anass 1109	. . . . 5 ⊢ (((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
14	12, 13	imbitrrdi 255	. . . 4 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
15	14	imp 411	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
16	2, 3	ablodivdiv4 30847	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
17	15, 16	syldan 602	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
18	2, 3	grpodivcl 30832	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19	18	3expib 1138	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
20	19	anim1d 622	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
21		an4 668	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
22		3anass 1109	. . . . . 6 ⊢ (((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
23	20, 21, 22	3imtr4g 299	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
24	23	imp 411	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
25	2, 3	grpomuldivass 30834	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
26	24, 25	syldan 602	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
27	8, 26	sylan 591	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
28	7, 17, 27	3eqtr3d 2812	1 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5663  ‘cfv 6537  (class class class)co 7411  GrpOpcgr 30782   /_𝑔 cgs 30785  AbelOpcablo 30837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-grpo 30786  df-gid 30787  df-ginv 30788  df-gdiv 30789  df-ablo 30838

This theorem is referenced by: (None)