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Theorem shjcom 22709
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 22702 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( _|_ `  ( _|_ `  ( A  u.  B )
) ) )
2 shjval 22702 . . . 4  |-  ( ( B  e.  SH  /\  A  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( B  u.  A )
) ) )
32ancoms 440 . . 3  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( B  u.  A )
) ) )
4 uncom 3435 . . . . 5  |-  ( B  u.  A )  =  ( A  u.  B
)
54fveq2i 5672 . . . 4  |-  ( _|_ `  ( B  u.  A
) )  =  ( _|_ `  ( A  u.  B ) )
65fveq2i 5672 . . 3  |-  ( _|_ `  ( _|_ `  ( B  u.  A )
) )  =  ( _|_ `  ( _|_ `  ( A  u.  B
) ) )
73, 6syl6eq 2436 . 2  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( B  vH  A
)  =  ( _|_ `  ( _|_ `  ( A  u.  B )
) ) )
81, 7eqtr4d 2423 1  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B
)  =  ( B  vH  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    u. cun 3262   ` cfv 5395  (class class class)co 6021   SHcsh 22280   _|_cort 22282    vH chj 22285
This theorem is referenced by:  shlej2  22712  shjcomi  22722  shub2  22734  chjcom  22857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-hilex 22351
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-sh 22558  df-chj 22661
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