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Theorem chjcom 23008
Description: Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chjcom  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )

Proof of Theorem chjcom
StepHypRef Expression
1 chsh 22727 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 chsh 22727 . 2  |-  ( B  e.  CH  ->  B  e.  SH )
3 shjcom 22860 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
41, 2, 3syl2an 464 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6081   SHcsh 22431   CHcch 22432    vH chj 22436
This theorem is referenced by:  chub2  23010  chlejb2  23015  chj12  23036  mddmd2  23812  dmdsl3  23818  csmdsymi  23837  mdexchi  23838  atordi  23887  atcvatlem  23888  atcvati  23889  chirredlem2  23894  chirredlem4  23896  atcvat3i  23899  atcvat4i  23900  atdmd  23901  mdsymlem3  23908  mdsymlem5  23910  mdsymlem8  23913  sumdmdlem2  23922  dmdbr5ati  23925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-sh 22709  df-ch 22724  df-chj 22812
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