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Theorem ablo4 27744
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 764 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐴𝑋)
2 simprlr 765 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
3 simprrl 766 . . . . . 6 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
41, 2, 33jca 1122 . . . . 5 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝑋𝐵𝑋𝐶𝑋))
5 ablcom.1 . . . . . 6 𝑋 = ran 𝐺
65ablo32 27743 . . . . 5 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
74, 6syldan 579 . . . 4 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
87oveq1d 6808 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷))
9 ablogrpo 27741 . . . 4 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
105grpocl 27694 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
11103expb 1113 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
1211adantrr 696 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐺𝐵) ∈ 𝑋)
13 simprrl 766 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
14 simprrr 767 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐷𝑋)
1512, 13, 143jca 1122 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋))
165grpoass 27697 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋)) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
1715, 16syldan 579 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
189, 17sylan 569 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐵)𝐺𝐶)𝐺𝐷) = ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)))
195grpocl 27694 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐺𝐶) ∈ 𝑋)
20193expb 1113 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
2120adantrlr 702 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → (𝐴𝐺𝐶) ∈ 𝑋)
2221adantrrr 704 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐺𝐶) ∈ 𝑋)
23 simprlr 765 . . . . . 6 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
2422, 23, 143jca 1122 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋))
255grpoass 27697 . . . . 5 ((𝐺 ∈ GrpOp ∧ ((𝐴𝐺𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋)) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
2624, 25syldan 579 . . . 4 ((𝐺 ∈ GrpOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
279, 26sylan 569 . . 3 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐺𝐶)𝐺𝐵)𝐺𝐷) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
288, 18, 273eqtr3d 2813 . 2 ((𝐺 ∈ AbelOp ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
29283impb 1107 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  (class class class)co 6793  GrpOpcgr 27683  AbelOpcablo 27738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-ov 6796  df-grpo 27687  df-ablo 27739
This theorem is referenced by:  nvadd4  27820  ipdirilem  28024  rngoa4  34047
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