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Theorem orthcom 26596
Description: Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
orthcom  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 ) )

Proof of Theorem orthcom
StepHypRef Expression
1 fveq2 5881 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
* `  ( A  .ih  B ) )  =  ( * `  0
) )
2 cj0 13200 . . . 4  |-  ( * `
 0 )  =  0
31, 2syl6eq 2486 . . 3  |-  ( ( A  .ih  B )  =  0  ->  (
* `  ( A  .ih  B ) )  =  0 )
4 ax-his1 26570 . . . . 5  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  =  ( * `
 ( A  .ih  B ) ) )
54ancoms 454 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( B  .ih  A
)  =  ( * `
 ( A  .ih  B ) ) )
65eqeq1d 2431 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( B  .ih  A )  =  0  <->  (
* `  ( A  .ih  B ) )  =  0 ) )
73, 6syl5ibr 224 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( B  .ih  A
)  =  0 ) )
8 fveq2 5881 . . . 4  |-  ( ( B  .ih  A )  =  0  ->  (
* `  ( B  .ih  A ) )  =  ( * `  0
) )
98, 2syl6eq 2486 . . 3  |-  ( ( B  .ih  A )  =  0  ->  (
* `  ( B  .ih  A ) )  =  0 )
10 ax-his1 26570 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )
1110eqeq1d 2431 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  (
* `  ( B  .ih  A ) )  =  0 ) )
129, 11syl5ibr 224 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( B  .ih  A )  =  0  -> 
( A  .ih  B
)  =  0 ) )
137, 12impbid 193 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   0cc0 9538   *ccj 13138   ~Hchil 26407    .ih csp 26410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-his1 26570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-2 10668  df-cj 13141  df-re 13142  df-im 13143
This theorem is referenced by:  normpythi  26630  ocorth  26779  shorth  26783  h1dei  27038  h1de2i  27041  pjspansn  27065
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