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Theorem shlej1 21939
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 3344 . . . 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 21789 . . . . . . 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 21789 . . . . . . 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 3351 . . . . 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 21789 . . . . . . 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 3351 . . . . 5  |-  ( ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( B  u.  C )  C_  ~H )
14 occon2 21867 . . . . 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 21930 . . 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 21930 . . 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 3221 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 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   ` cfv 5255  (class class class)co 5858   ~Hchil 21499   SHcsh 21508   _|_cort 21510    vH chj 21513
This theorem is referenced by:  shlej2  21940  shlej1i  21957  chlej1  22089
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585  ax-hvmul0 21590  ax-hfi 21658  ax-his2 21662  ax-his3 21663
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 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-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sh 21786  df-oc 21831  df-chj 21889
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