Theorem ablo4pnp 38815

Description: Obsolete theorem, use ablsub4 20024 instead. A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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 1105	. . . . 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 31154	. . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
5	1, 4	sylan2br 607	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
6	5	adantrrr 738	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
7	6	oveq1d 7435	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8		ablogrpo 31149	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
9	2	grpocl 31102	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
10	9	3expib 1140	. . . . . . 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 623	. . . . 5 ⊢ (𝐺 ∈ AbelOp → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
13		3anass 1111	. . . . 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 412	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
16	2, 3	ablodivdiv4 31156	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
17	15, 16	syldan 603	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
18	2, 3	grpodivcl 31141	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19	18	3expib 1140	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
20	19	anim1d 623	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))))
21		an4 669	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)))
22		3anass 1111	. . . . . 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 412	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))
25	2, 3	grpomuldivass 31143	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
26	24, 25	syldan 603	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
27	8, 26	sylan 592	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
28	7, 17, 27	3eqtr3d 2803	1 ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ran crn 5651  ‘cfv 6538  (class class class)co 7420  GrpOpcgr 31091   /_𝑔 cgs 31094  AbelOpcablo 31146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-grpo 31095  df-gid 31096  df-ginv 31097  df-gdiv 31098  df-ablo 31147

This theorem is used by: (None)