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Theorem shlej1 26404
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 3669 . . . 4  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
3 simpl1 999 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  A  e.  SH )
4 shss 26253 . . . . . . 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 1001 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  C  e.  SH )
7 shss 26253 . . . . . . 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 3676 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  u.  C )  C_  ~H )
10 simpl2 1000 . . . . . . 7  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  B  e.  SH )
11 shss 26253 . . . . . . 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 3676 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  u.  C )  C_  ~H )
14 occon2 26332 . . . . 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 26395 . . 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 26395 . . 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 3542 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 973    = wceq 1395    e. wcel 1819    u. cun 3469    C_ wss 3471   ` cfv 5594  (class class class)co 6296   ~Hchil 25962   SHcsh 25971   _|_cort 25973    vH chj 25976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-hilex 26042  ax-hfvadd 26043  ax-hv0cl 26046  ax-hfvmul 26048  ax-hvmul0 26053  ax-hfi 26122  ax-his2 26126  ax-his3 26127
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sh 26250  df-oc 26296  df-chj 26354
This theorem is referenced by:  shlej2  26405  shlej1i  26422  chlej1  26554
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