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Theorem ablodiv32 29539
Description: Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1 𝑋 = ran 𝐺
abldiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
ablodiv32 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵))

Proof of Theorem ablodiv32
StepHypRef Expression
1 abldiv.1 . . . . 5 𝑋 = ran 𝐺
21ablocom 29532 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
323adant3r1 1183 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
43oveq2d 7374 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐺𝐶)) = (𝐴𝐷(𝐶𝐺𝐵)))
5 abldiv.3 . . 3 𝐷 = ( /𝑔𝐺)
61, 5ablodivdiv4 29538 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
7 3ancomb 1100 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ (𝐴𝑋𝐶𝑋𝐵𝑋))
81, 5ablodivdiv4 29538 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → ((𝐴𝐷𝐶)𝐷𝐵) = (𝐴𝐷(𝐶𝐺𝐵)))
97, 8sylan2b 595 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐶)𝐷𝐵) = (𝐴𝐷(𝐶𝐺𝐵)))
104, 6, 93eqtr4d 2783 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  ran crn 5635  cfv 6497  (class class class)co 7358   /𝑔 cgs 29476  AbelOpcablo 29528
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480  df-ablo 29529
This theorem is referenced by:  ablonnncan1  29541
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