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Theorem shlej1 21955
Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
shlej1  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )

Proof of Theorem shlej1
StepHypRef Expression
1 simpr 447 . . 3  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  C_  B
)
2 unss1 3357 . . . 4  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
3 simpl1 958 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  e.  SH )
4 shss 21805 . . . . . . 7  |-  ( A  e.  SH  ->  A  C_ 
~H )
53, 4syl 15 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  C_  ~H )
6 simpl3 960 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  C  e.  SH )
7 shss 21805 . . . . . . 7  |-  ( C  e.  SH  ->  C  C_ 
~H )
86, 7syl 15 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  C  C_  ~H )
95, 8unssd 3364 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  u.  C )  C_  ~H )
10 simpl2 959 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  B  e.  SH )
11 shss 21805 . . . . . . 7  |-  ( B  e.  SH  ->  B  C_ 
~H )
1210, 11syl 15 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  B  C_  ~H )
1312, 8unssd 3364 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  u.  C )  C_  ~H )
14 occon2 21883 . . . . 5  |-  ( ( ( A  u.  C
)  C_  ~H  /\  ( B  u.  C )  C_ 
~H )  ->  (
( A  u.  C
)  C_  ( B  u.  C )  ->  ( _|_ `  ( _|_ `  ( A  u.  C )
) )  C_  ( _|_ `  ( _|_ `  ( B  u.  C )
) ) ) )
159, 13, 14syl2anc 642 . . . 4  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( ( A  u.  C )  C_  ( B  u.  C
)  ->  ( _|_ `  ( _|_ `  ( A  u.  C )
) )  C_  ( _|_ `  ( _|_ `  ( B  u.  C )
) ) ) )
162, 15syl5 28 . . 3  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  C_  B  ->  ( _|_ `  ( _|_ `  ( A  u.  C ) ) ) 
C_  ( _|_ `  ( _|_ `  ( B  u.  C ) ) ) ) )
171, 16mpd 14 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( _|_ `  ( _|_ `  ( A  u.  C ) ) ) 
C_  ( _|_ `  ( _|_ `  ( B  u.  C ) ) ) )
18 shjval 21946 . . 3  |-  ( ( A  e.  SH  /\  C  e.  SH )  ->  ( A  vH  C
)  =  ( _|_ `  ( _|_ `  ( A  u.  C )
) ) )
193, 6, 18syl2anc 642 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  =  ( _|_ `  ( _|_ `  ( A  u.  C
) ) ) )
20 shjval 21946 . . 3  |-  ( ( B  e.  SH  /\  C  e.  SH )  ->  ( B  vH  C
)  =  ( _|_ `  ( _|_ `  ( B  u.  C )
) ) )
2110, 6, 20syl2anc 642 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  vH  C )  =  ( _|_ `  ( _|_ `  ( B  u.  C
) ) ) )
2217, 19, 213sstr4d 3234 1  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   ` cfv 5271  (class class class)co 5874   ~Hchil 21515   SHcsh 21524   _|_cort 21526    vH chj 21529
This theorem is referenced by:  shlej2  21956  shlej1i  21973  chlej1  22105
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmul0 21606  ax-hfi 21674  ax-his2 21678  ax-his3 21679
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sh 21802  df-oc 21847  df-chj 21905
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