Theorem ablo4pnp 38384

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 1101	. . . . 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 30757	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
5	1, 4	sylan2br 604	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
6	5	adantrrr 735	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
7	6	oveq1d 7413	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8		ablogrpo 30752	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9	2	grpocl 30705	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
10	9	3expib 1136	. . . . . . 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 620	. . . . 5 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
13		3anass 1107	. . . . 5 ⊢ (((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
14	12, 13	imbitrrdi 254	. . . 4 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
15	14	imp 410	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
16	2, 3	ablodivdiv4 30759	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
17	15, 16	syldan 600	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
18	2, 3	grpodivcl 30744	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19	18	3expib 1136	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
20	19	anim1d 620	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
21		an4 666	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
22		3anass 1107	. . . . . 6 ⊢ (((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
23	20, 21, 22	3imtr4g 298	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
24	23	imp 410	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
25	2, 3	grpomuldivass 30746	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
26	24, 25	syldan 600	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
27	8, 26	sylan 589	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
28	7, 17, 27	3eqtr3d 2807	1 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ran crn 5650  ‘cfv 6523  (class class class)co 7398  GrpOpcgr 30694   /_𝑔 cgs 30697  AbelOpcablo 30749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-grpo 30698  df-gid 30699  df-ginv 30700  df-gdiv 30701  df-ablo 30750

This theorem is referenced by: (None)