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Theorem shorth 26414
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 3483 . . . . . 6  |-  ( G 
C_  ( _|_ `  H
)  ->  ( A  e.  G  ->  A  e.  ( _|_ `  H
) ) )
21anim1d 562 . . . . 5  |-  ( G 
C_  ( _|_ `  H
)  ->  ( ( A  e.  G  /\  B  e.  H )  ->  ( A  e.  ( _|_ `  H )  /\  B  e.  H
) ) )
32imp 427 . . . 4  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( A  e.  ( _|_ `  H
)  /\  B  e.  H ) )
43ancomd 449 . . 3  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( B  e.  H  /\  A  e.  ( _|_ `  H
) ) )
5 shocorth 26411 . . . . 5  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  .ih  A )  =  0 ) )
65imp 427 . . . 4  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  .ih  A )  =  0 )
7 shss 26328 . . . . . . . 8  |-  ( H  e.  SH  ->  H  C_ 
~H )
87sseld 3488 . . . . . . 7  |-  ( H  e.  SH  ->  ( B  e.  H  ->  B  e.  ~H ) )
9 shocss 26405 . . . . . . . 8  |-  ( H  e.  SH  ->  ( _|_ `  H )  C_  ~H )
109sseld 3488 . . . . . . 7  |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  ->  A  e.  ~H ) )
118, 10anim12d 561 . . . . . 6  |-  ( H  e.  SH  ->  (
( B  e.  H  /\  A  e.  ( _|_ `  H ) )  ->  ( B  e. 
~H  /\  A  e.  ~H ) ) )
1211imp 427 . . . . 5  |-  ( ( H  e.  SH  /\  ( B  e.  H  /\  A  e.  ( _|_ `  H ) ) )  ->  ( B  e.  ~H  /\  A  e. 
~H ) )
13 orthcom 26226 . . . . 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 472 . 2  |-  ( ( H  e.  SH  /\  ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
) )  ->  ( A  .ih  B )  =  0 )
1716exp32 603 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570  (class class class)co 6270   0cc0 9481   ~Hchil 26037    .ih csp 26040   SHcsh 26046   _|_cort 26048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-hilex 26117  ax-hfvadd 26118  ax-hv0cl 26121  ax-hfvmul 26123  ax-hvmul0 26128  ax-hfi 26197  ax-his1 26200  ax-his2 26201  ax-his3 26202
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-cj 13017  df-re 13018  df-im 13019  df-sh 26325  df-oc 26371
This theorem is referenced by:  pjoi0  26836
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