HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chjcom Structured version   Visualization version   GIF version

Theorem chjcom 28695
Description: Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chjcom ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))

Proof of Theorem chjcom
StepHypRef Expression
1 chsh 28411 . 2 (𝐴C𝐴S )
2 chsh 28411 . 2 (𝐵C𝐵S )
3 shjcom 28547 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
41, 2, 3syl2an 495 1 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  (class class class)co 6814   S csh 28115   C cch 28116   chj 28120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hilex 28186
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-sh 28394  df-ch 28408  df-chj 28499
This theorem is referenced by:  chub2  28697  chlejb2  28702  chj12  28723  mddmd2  29498  dmdsl3  29504  csmdsymi  29523  mdexchi  29524  atordi  29573  atcvatlem  29574  atcvati  29575  chirredlem2  29580  chirredlem4  29582  atcvat3i  29585  atcvat4i  29586  atdmd  29587  mdsymlem3  29594  mdsymlem5  29596  mdsymlem8  29599  sumdmdlem2  29608  dmdbr5ati  29611
  Copyright terms: Public domain W3C validator