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Theorem ablo4pnp 33346
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 = ( /𝑔𝐺)
Assertion
Ref Expression
ablo4pnp ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))

Proof of Theorem ablo4pnp
StepHypRef Expression
1 df-3an 1038 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋))
2 abl4pnp.1 . . . . . 6 𝑋 = ran 𝐺
3 abl4pnp.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
42, 3ablomuldiv 27276 . . . . 5 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
51, 4sylan2br 493 . . . 4 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
65adantrrr 760 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
76oveq1d 6625 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8 ablogrpo 27271 . . . . . . 7 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
92grpocl 27224 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
1093expib 1265 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
118, 10syl 17 . . . . . 6 (𝐺 ∈ AbelOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
1211anim1d 587 . . . . 5 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋))))
13 3anass 1040 . . . . 5 (((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋)))
1412, 13syl6ibr 242 . . . 4 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)))
1514imp 445 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋))
162, 3ablodivdiv4 27278 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
1715, 16syldan 487 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
182, 3grpodivcl 27263 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19183expib 1265 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
2019anim1d 587 . . . . . 6 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋))))
21 an4 864 . . . . . 6 (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) ↔ ((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)))
22 3anass 1040 . . . . . 6 (((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋)))
2320, 21, 223imtr4g 285 . . . . 5 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)))
2423imp 445 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋))
252, 3grpomuldivass 27265 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
2624, 25syldan 487 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
278, 26sylan 488 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
287, 17, 273eqtr3d 2663 1 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ran crn 5080  cfv 5852  (class class class)co 6610  GrpOpcgr 27213   /𝑔 cgs 27216  AbelOpcablo 27268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-grpo 27217  df-gid 27218  df-ginv 27219  df-gdiv 27220  df-ablo 27269
This theorem is referenced by: (None)
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