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Theorem chjcom 22085
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 21804 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 chsh 21804 . 2  |-  ( B  e.  CH  ->  B  e.  SH )
3 shjcom 21937 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
41, 2, 3syl2an 463 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   SHcsh 21508   CHcch 21509    vH chj 21513
This theorem is referenced by:  chub2  22087  chlejb2  22092  chj12  22113  mddmd2  22889  dmdsl3  22895  csmdsymi  22914  mdexchi  22915  atordi  22964  atcvatlem  22965  atcvati  22966  chirredlem2  22971  chirredlem4  22973  atcvat3i  22976  atcvat4i  22977  atdmd  22978  mdsymlem3  22985  mdsymlem5  22987  mdsymlem8  22990  sumdmdlem2  22999  dmdbr5ati  23002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sh 21786  df-ch 21801  df-chj 21889
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