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Theorem shorth 22802
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 3344 . . . . . 6  |-  ( G 
C_  ( _|_ `  H
)  ->  ( A  e.  G  ->  A  e.  ( _|_ `  H
) ) )
21anim1d 549 . . . . 5  |-  ( G 
C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  e.  ( _|_ `  H )  /\  B  e.  H
) ) )
32imp 420 . . . 4  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( A  e.  ( _|_ `  H
)  /\  B  e.  H ) )
43ancomd 440 . . 3  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( B  e.  H  /\  A  e.  ( _|_ `  H
) ) )
5 shocorth 22799 . . . . 5  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  .ih  A )  =  0 ) )
65imp 420 . . . 4  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  .ih  A )  =  0 )
7 shss 22717 . . . . . . . 8  |-  ( H  e.  SH  ->  H  C_ 
~H )
87sseld 3349 . . . . . . 7  |-  ( H  e.  SH  ->  ( B  e.  H  ->  B  e.  ~H ) )
9 shocss 22793 . . . . . . . 8  |-  ( H  e.  SH  ->  ( _|_ `  H )  C_  ~H )
109sseld 3349 . . . . . . 7  |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  ->  A  e.  ~H ) )
118, 10anim12d 548 . . . . . 6  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  e. 
~H  /\  A  e.  ~H ) ) )
1211imp 420 . . . . 5  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  e.  ~H  /\  A  e. 
~H ) )
13 orthcom 22615 . . . . 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 203 . . 3  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( A  .ih  B )  =  0 )
164, 15sylan2 462 . 2  |-  ( ( H  e.  SH  /\  ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
) )  ->  ( A  .ih  B )  =  0 )
1716exp32 590 1  |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  .ih  B
)  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5457  (class class class)co 6084   0cc0 8995   ~Hchil 22427    .ih csp 22430   SHcsh 22436   _|_cort 22438
This theorem is referenced by:  pjoi0  23224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-hilex 22507  ax-hfvadd 22508  ax-hv0cl 22511  ax-hfvmul 22513  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-cj 11909  df-re 11910  df-im 11911  df-sh 22714  df-oc 22759
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