Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngocom Structured version   Visualization version   GIF version

Theorem rngocom 34044
Description: The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringgcl.1 𝐺 = (1st𝑅)
ringgcl.2 𝑋 = ran 𝐺
Assertion
Ref Expression
rngocom ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Proof of Theorem rngocom
StepHypRef Expression
1 ringgcl.1 . . 3 𝐺 = (1st𝑅)
21rngoablo 34039 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ringgcl.2 . . 3 𝑋 = ran 𝐺
43ablocom 27742 . 2 ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
52, 4syl3an1 1166 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  cfv 6031  (class class class)co 6793  1st c1st 7313  AbelOpcablo 27738  RingOpscrngo 34025
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-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-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-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-fv 6039  df-ov 6796  df-1st 7315  df-2nd 7316  df-ablo 27739  df-rngo 34026
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator