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Theorem chjcom 29267
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 28985 . 2 (𝐴C𝐴S )
2 chsh 28985 . 2 (𝐵C𝐵S )
3 shjcom 29119 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
41, 2, 3syl2an 598 1 ((𝐴C𝐵C ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  (class class class)co 7130   S csh 28689   C cch 28690   chj 28694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-hilex 28760
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-sh 28968  df-ch 28982  df-chj 29071
This theorem is referenced by:  chub2  29269  chlejb2  29274  chj12  29295  mddmd2  30070  dmdsl3  30076  csmdsymi  30095  mdexchi  30096  atordi  30145  atcvatlem  30146  atcvati  30147  chirredlem2  30152  chirredlem4  30154  atcvat3i  30157  atcvat4i  30158  atdmd  30159  mdsymlem3  30166  mdsymlem5  30168  mdsymlem8  30171  sumdmdlem2  30180  dmdbr5ati  30183
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