HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shjcom Unicode version

Theorem shjcom 21937
Description: Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shjcom  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )

Proof of Theorem shjcom
StepHypRef Expression
1 shjval 21930 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( _|_ `  ( _|_ `  ( A  u.  B )
) ) )
2 shjval 21930 . . . 4  |-  ( ( B  e.  SH  /\  A  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( B  u.  A )
) ) )
32ancoms 439 . . 3  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( B  u.  A )
) ) )
4 uncom 3319 . . . . 5  |-  ( B  u.  A )  =  ( A  u.  B
)
54fveq2i 5528 . . . 4  |-  ( _|_ `  ( B  u.  A
) )  =  ( _|_ `  ( A  u.  B ) )
65fveq2i 5528 . . 3  |-  ( _|_ `  ( _|_ `  ( B  u.  A )
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) )
73, 6syl6eq 2331 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( A  u.  B )
) ) )
81, 7eqtr4d 2318 1  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   ` cfv 5255  (class class class)co 5858   SHcsh 21508   _|_cort 21510    vH chj 21513
This theorem is referenced by:  shlej2  21940  shjcomi  21950  shub2  21962  chjcom  22085
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-chj 21889
  Copyright terms: Public domain W3C validator