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Theorem shorth 24521
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 3338 . . . . . 6  |-  ( G 
C_  ( _|_ `  H
)  ->  ( A  e.  G  ->  A  e.  ( _|_ `  H
) ) )
21anim1d 559 . . . . 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 449 . . 3  |-  ( ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
)  ->  ( B  e.  H  /\  A  e.  ( _|_ `  H
) ) )
5 shocorth 24518 . . . . 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 24435 . . . . . . . 8  |-  ( H  e.  SH  ->  H  C_ 
~H )
87sseld 3343 . . . . . . 7  |-  ( H  e.  SH  ->  ( B  e.  H  ->  B  e.  ~H ) )
9 shocss 24512 . . . . . . . 8  |-  ( H  e.  SH  ->  ( _|_ `  H )  C_  ~H )
109sseld 3343 . . . . . . 7  |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  ->  A  e.  ~H ) )
118, 10anim12d 558 . . . . . 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 24333 . . . . 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 471 . 2  |-  ( ( H  e.  SH  /\  ( G  C_  ( _|_ `  H )  /\  ( A  e.  G  /\  B  e.  H )
) )  ->  ( A  .ih  B )  =  0 )
1716exp32 600 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 1362    e. wcel 1755    C_ wss 3316   ` cfv 5406  (class class class)co 6080   0cc0 9270   ~Hchil 24144    .ih csp 24147   SHcsh 24153   _|_cort 24155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-hilex 24224  ax-hfvadd 24225  ax-hv0cl 24228  ax-hfvmul 24230  ax-hvmul0 24235  ax-hfi 24304  ax-his1 24307  ax-his2 24308  ax-his3 24309
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-2 10368  df-cj 12572  df-re 12573  df-im 12574  df-sh 24432  df-oc 24478
This theorem is referenced by:  pjoi0  24943
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