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Theorem chjcom 22961
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 22680 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 chsh 22680 . 2  |-  ( B  e.  CH  ->  B  e.  SH )
3 shjcom 22813 . 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 1649    e. wcel 1721  (class class class)co 6040   SHcsh 22384   CHcch 22385    vH chj 22389
This theorem is referenced by:  chub2  22963  chlejb2  22968  chj12  22989  mddmd2  23765  dmdsl3  23771  csmdsymi  23790  mdexchi  23791  atordi  23840  atcvatlem  23841  atcvati  23842  chirredlem2  23847  chirredlem4  23849  atcvat3i  23852  atcvat4i  23853  atdmd  23854  mdsymlem3  23861  mdsymlem5  23863  mdsymlem8  23866  sumdmdlem2  23875  dmdbr5ati  23878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-hilex 22455
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-sh 22662  df-ch 22677  df-chj 22765
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