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Theorem ablo4pnp 36035
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 1088 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋))
2 abl4pnp.1 . . . . . 6 𝑋 = ran 𝐺
3 abl4pnp.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
42, 3ablomuldiv 28911 . . . . 5 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
51, 4sylan2br 595 . . . 4 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
65adantrrr 722 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
76oveq1d 7292 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8 ablogrpo 28906 . . . . . . 7 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
92grpocl 28859 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
1093expib 1121 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
118, 10syl 17 . . . . . 6 (𝐺 ∈ AbelOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
1211anim1d 611 . . . . 5 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋))))
13 3anass 1094 . . . . 5 (((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋)))
1412, 13syl6ibr 251 . . . 4 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)))
1514imp 407 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋))
162, 3ablodivdiv4 28913 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
1715, 16syldan 591 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
182, 3grpodivcl 28898 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19183expib 1121 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
2019anim1d 611 . . . . . 6 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋))))
21 an4 653 . . . . . 6 (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) ↔ ((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)))
22 3anass 1094 . . . . . 6 (((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋)))
2320, 21, 223imtr4g 296 . . . . 5 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)))
2423imp 407 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋))
252, 3grpomuldivass 28900 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
2624, 25syldan 591 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
278, 26sylan 580 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
287, 17, 273eqtr3d 2786 1 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  ran crn 5592  cfv 6435  (class class class)co 7277  GrpOpcgr 28848   /𝑔 cgs 28851  AbelOpcablo 28903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-1st 7831  df-2nd 7832  df-grpo 28852  df-gid 28853  df-ginv 28854  df-gdiv 28855  df-ablo 28904
This theorem is referenced by: (None)
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