HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shlej1 Structured version   Unicode version

Theorem shlej1 24761
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 461 . . 3  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  C_  B
)
2 unss1 3523 . . . 4  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
3 simpl1 991 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  e.  SH )
4 shss 24610 . . . . . . 7  |-  ( A  e.  SH  ->  A  C_ 
~H )
53, 4syl 16 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  C_  ~H )
6 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  C  e.  SH )
7 shss 24610 . . . . . . 7  |-  ( C  e.  SH  ->  C  C_ 
~H )
86, 7syl 16 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  C  C_  ~H )
95, 8unssd 3530 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  u.  C )  C_  ~H )
10 simpl2 992 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  B  e.  SH )
11 shss 24610 . . . . . . 7  |-  ( B  e.  SH  ->  B  C_ 
~H )
1210, 11syl 16 . . . . . 6  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  B  C_  ~H )
1312, 8unssd 3530 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  u.  C )  C_  ~H )
14 occon2 24689 . . . . 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 661 . . . 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 32 . . 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 15 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( _|_ `  ( _|_ `  ( A  u.  C ) ) ) 
C_  ( _|_ `  ( _|_ `  ( B  u.  C ) ) ) )
18 shjval 24752 . . 3  |-  ( ( A  e.  SH  /\  C  e.  SH )  ->  ( A  vH  C
)  =  ( _|_ `  ( _|_ `  ( A  u.  C )
) ) )
193, 6, 18syl2anc 661 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  =  ( _|_ `  ( _|_ `  ( A  u.  C
) ) ) )
20 shjval 24752 . . 3  |-  ( ( B  e.  SH  /\  C  e.  SH )  ->  ( B  vH  C
)  =  ( _|_ `  ( _|_ `  ( B  u.  C )
) ) )
2110, 6, 20syl2anc 661 . 2  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  vH  C )  =  ( _|_ `  ( _|_ `  ( B  u.  C
) ) ) )
2217, 19, 213sstr4d 3397 1  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3324    C_ wss 3326   ` cfv 5416  (class class class)co 6089   ~Hchil 24319   SHcsh 24328   _|_cort 24330    vH chj 24333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-hilex 24399  ax-hfvadd 24400  ax-hv0cl 24403  ax-hfvmul 24405  ax-hvmul0 24410  ax-hfi 24479  ax-his2 24483  ax-his3 24484
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-ltxr 9421  df-sh 24607  df-oc 24653  df-chj 24711
This theorem is referenced by:  shlej2  24762  shlej1i  24779  chlej1  24911
  Copyright terms: Public domain W3C validator