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Theorem ablo4 30478
Description: Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablcom.1 𝑋 = ran 𝐺
Assertion
Ref Expression
ablo4 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Proof of Theorem ablo4
StepHypRef Expression
1 simprll 777 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐴𝑋)
2 simprlr 778 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
3 simprrl 779 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
41, 2, 33jca 1125 . . . . 5 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝑋𝐵𝑋𝐶𝑋))
5 ablcom.1 . . . . . 6 𝑋 = ran 𝐺
65ablo32 30477 . . . . 5 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
74, 6syldan 589 . . . 4 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
87oveq1d 7429 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷))
9 ablogrpo 30475 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
105grpocl 30428 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
11103expb 1117 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
1211adantrr 715 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐺𝐵) ∈ 𝑋)
13 simprrl 779 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
14 simprrr 780 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐷𝑋)
1512, 13, 143jca 1125 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋))
165grpoass 30431 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋)) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
1715, 16syldan 589 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
189, 17sylan 578 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
195grpocl 30428 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐺𝐶) ∈ 𝑋)
20193expb 1117 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
2120adantrlr 721 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
2221adantrrr 723 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐺𝐶) ∈ 𝑋)
23 simprlr 778 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
2422, 23, 143jca 1125 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋))
255grpoass 30431 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋)) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
2624, 25syldan 589 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
279, 26sylan 578 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
288, 18, 273eqtr3d 2774 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
29283impb 1112 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  ran crn 5674  (class class class)co 7414  GrpOpcgr 30417  AbelOpcablo 30472
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 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7417  df-grpo 30421  df-ablo 30473
This theorem is referenced by:  nvadd4  30553  ipdirilem  30757  rngoa4  37628
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