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Theorem chjcom 22101
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 21820 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 chsh 21820 . 2  |-  ( B  e.  CH  ->  B  e.  SH )
3 shjcom 21953 . 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 1632    e. wcel 1696  (class class class)co 5874   SHcsh 21524   CHcch 21525    vH chj 21529
This theorem is referenced by:  chub2  22103  chlejb2  22108  chj12  22129  mddmd2  22905  dmdsl3  22911  csmdsymi  22930  mdexchi  22931  atordi  22980  atcvatlem  22981  atcvati  22982  chirredlem2  22987  chirredlem4  22989  atcvat3i  22992  atcvat4i  22993  atdmd  22994  mdsymlem3  23001  mdsymlem5  23003  mdsymlem8  23006  sumdmdlem2  23015  dmdbr5ati  23018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-sh 21802  df-ch 21817  df-chj 21905
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