MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablo32 Structured version   Visualization version   GIF version

Theorem ablo32 28103
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
ablo32 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Proof of Theorem ablo32
StepHypRef Expression
1 ablcom.1 . . . . 5 𝑋 = ran 𝐺
21ablocom 28102 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
323adant3r1 1162 . . 3 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
43oveq2d 6992 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺(𝐵𝐺𝐶)) = (𝐴𝐺(𝐶𝐺𝐵)))
5 ablogrpo 28101 . . 3 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
61grpoass 28057 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
75, 6sylan 572 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
8 3ancomb 1080 . . . 4 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ (𝐴𝑋𝐶𝑋𝐵𝑋))
91grpoass 28057 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → ((𝐴𝐺𝐶)𝐺𝐵) = (𝐴𝐺(𝐶𝐺𝐵)))
108, 9sylan2b 584 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶)𝐺𝐵) = (𝐴𝐺(𝐶𝐺𝐵)))
115, 10sylan 572 . 2 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶)𝐺𝐵) = (𝐴𝐺(𝐶𝐺𝐵)))
124, 7, 113eqtr4d 2824 1 ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  ran crn 5408  (class class class)co 6976  GrpOpcgr 28043  AbelOpcablo 28098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fo 6194  df-fv 6196  df-ov 6979  df-grpo 28047  df-ablo 28099
This theorem is referenced by:  ablo4  28104  nvadd32  28177  ip0i  28379  rngoa32  34641
  Copyright terms: Public domain W3C validator