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Theorem chjcom 27551
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 27267 . 2 (𝐴C𝐴S )
2 chsh 27267 . 2 (𝐵C𝐵S )
3 shjcom 27403 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
41, 2, 3syl2an 492 1 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  (class class class)co 6523   S csh 26971   C cch 26972   chj 26976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-hilex 27042
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-sh 27250  df-ch 27264  df-chj 27355
This theorem is referenced by:  chub2  27553  chlejb2  27558  chj12  27579  mddmd2  28354  dmdsl3  28360  csmdsymi  28379  mdexchi  28380  atordi  28429  atcvatlem  28430  atcvati  28431  chirredlem2  28436  chirredlem4  28438  atcvat3i  28441  atcvat4i  28442  atdmd  28443  mdsymlem3  28450  mdsymlem5  28452  mdsymlem8  28455  sumdmdlem2  28464  dmdbr5ati  28467
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