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Theorem ablo4pnp 37275
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 1087 . . . . 5 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋))
2 abl4pnp.1 . . . . . 6 𝑋 = ran 𝐺
3 abl4pnp.2 . . . . . 6 𝐷 = ( /𝑔𝐺)
42, 3ablomuldiv 30336 . . . . 5 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
51, 4sylan2br 594 . . . 4 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
65adantrrr 724 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
76oveq1d 7429 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹))
8 ablogrpo 30331 . . . . . . 7 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
92grpocl 30284 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
1093expib 1120 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
118, 10syl 17 . . . . . 6 (𝐺 ∈ AbelOp → ((𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋))
1211anim1d 610 . . . . 5 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋))))
13 3anass 1093 . . . . 5 (((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋) ↔ ((𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝑋𝐹𝑋)))
1412, 13imbitrrdi 251 . . . 4 (𝐺 ∈ AbelOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)))
1514imp 406 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋))
162, 3ablodivdiv4 30338 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐹𝑋)) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
1715, 16syldan 590 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐺𝐵)𝐷𝐶)𝐷𝐹) = ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)))
182, 3grpodivcl 30323 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋)
19183expib 1120 . . . . . . 7 (𝐺 ∈ GrpOp → ((𝐴𝑋𝐶𝑋) → (𝐴𝐷𝐶) ∈ 𝑋))
2019anim1d 610 . . . . . 6 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋))))
21 an4 655 . . . . . 6 (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) ↔ ((𝐴𝑋𝐶𝑋) ∧ (𝐵𝑋𝐹𝑋)))
22 3anass 1093 . . . . . 6 (((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋) ↔ ((𝐴𝐷𝐶) ∈ 𝑋 ∧ (𝐵𝑋𝐹𝑋)))
2320, 21, 223imtr4g 296 . . . . 5 (𝐺 ∈ GrpOp → (((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋)) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)))
2423imp 406 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋))
252, 3grpomuldivass 30325 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐷𝐶) ∈ 𝑋𝐵𝑋𝐹𝑋)) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
2624, 25syldan 590 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
278, 26sylan 579 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → (((𝐴𝐷𝐶)𝐺𝐵)𝐷𝐹) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
287, 17, 273eqtr3d 2775 1 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐹𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  ran crn 5673  cfv 6542  (class class class)co 7414  GrpOpcgr 30273   /𝑔 cgs 30276  AbelOpcablo 30328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-grpo 30277  df-gid 30278  df-ginv 30279  df-gdiv 30280  df-ablo 30329
This theorem is referenced by: (None)
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