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Theorem shorth 24649
Description: Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shorth  |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  .ih  B
)  =  0 ) ) )

Proof of Theorem shorth
StepHypRef Expression
1 ssel 3345 . . . . . 6  |-  ( G 
C_  ( _|_ `  H
)  ->  ( A  e.  G  ->  A  e.  ( _|_ `  H
) ) )
21anim1d 564 . . . . 5  |-  ( G 
C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  e.  ( _|_ `  H )  /\  B  e.  H
) ) )
32imp 429 . . . 4  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( A  e.  ( _|_ `  H
)  /\  B  e.  H ) )
43ancomd 451 . . 3  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( B  e.  H  /\  A  e.  ( _|_ `  H
) ) )
5 shocorth 24646 . . . . 5  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  .ih  A )  =  0 ) )
65imp 429 . . . 4  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  .ih  A )  =  0 )
7 shss 24563 . . . . . . . 8  |-  ( H  e.  SH  ->  H  C_ 
~H )
87sseld 3350 . . . . . . 7  |-  ( H  e.  SH  ->  ( B  e.  H  ->  B  e.  ~H ) )
9 shocss 24640 . . . . . . . 8  |-  ( H  e.  SH  ->  ( _|_ `  H )  C_  ~H )
109sseld 3350 . . . . . . 7  |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  ->  A  e.  ~H ) )
118, 10anim12d 563 . . . . . 6  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  e. 
~H  /\  A  e.  ~H ) ) )
1211imp 429 . . . . 5  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  e.  ~H  /\  A  e. 
~H ) )
13 orthcom 24461 . . . . 5  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( ( B  .ih  A )  =  0  <->  ( A  .ih  B )  =  0 ) )
1412, 13syl 16 . . . 4  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( ( B  .ih  A )  =  0  <->  ( A  .ih  B )  =  0 ) )
156, 14mpbid 210 . . 3  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( A  .ih  B )  =  0 )
164, 15sylan2 474 . 2  |-  ( ( H  e.  SH  /\  ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
) )  ->  ( A  .ih  B )  =  0 )
1716exp32 605 1  |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  .ih  B
)  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   ` cfv 5413  (class class class)co 6086   0cc0 9274   ~Hchil 24272    .ih csp 24275   SHcsh 24281   _|_cort 24283
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-hilex 24352  ax-hfvadd 24353  ax-hv0cl 24356  ax-hfvmul 24358  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-cj 12580  df-re 12581  df-im 12582  df-sh 24560  df-oc 24606
This theorem is referenced by:  pjoi0  25071
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