HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  normpyth Unicode version

Theorem normpyth 22600
Description: Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
normpyth  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( (
normh `  B ) ^
2 ) ) ) )

Proof of Theorem normpyth
StepHypRef Expression
1 oveq1 6047 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  .ih  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B
) )
21eqeq1d 2412 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  .ih  B
)  =  0  <->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  0 ) )
3 oveq1 6047 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
43fveq2d 5691 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
54oveq1d 6055 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  ( A  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 ) )
6 fveq2 5687 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  A )  =  ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) )
76oveq1d 6055 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  A ) ^ 2 )  =  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )
87oveq1d 6055 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  A
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) )
95, 8eqeq12d 2418 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( (
normh `  B ) ^
2 ) )  <->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) ) )
102, 9imbi12d 312 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  .ih  B )  =  0  -> 
( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( (
normh `  B ) ^
2 ) ) )  <-> 
( ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  0  ->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 )  +  ( ( normh `  B ) ^ 2 ) ) ) ) )
11 oveq2 6048 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  .ih  if ( B  e.  ~H ,  B ,  0h )
) )
1211eqeq1d 2412 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  B
)  =  0  <->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  if ( B  e. 
~H ,  B ,  0h ) )  =  0 ) )
13 oveq2 6048 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) )
1413fveq2d 5691 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) )
1514oveq1d 6055 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  =  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 ) )
16 fveq2 5687 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  B )  =  ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) )
1716oveq1d 6055 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  B ) ^ 2 )  =  ( ( normh `  if ( B  e.  ~H ,  B ,  0h )
) ^ 2 ) )
1817oveq2d 6056 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  if ( B  e. 
~H ,  B ,  0h ) ) ^ 2 ) ) )
1915, 18eqeq12d 2418 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) ^
2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) )  <->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  if ( B  e. 
~H ,  B ,  0h ) ) ^ 2 ) ) ) )
2012, 19imbi12d 312 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  0  ->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) ) ^ 2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 )  +  ( ( normh `  B ) ^ 2 ) ) )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  .ih  if ( B  e. 
~H ,  B ,  0h ) )  =  0  ->  ( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) ) ^ 2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) ^
2 )  +  ( ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) ^
2 ) ) ) ) )
21 ax-hv0cl 22459 . . . 4  |-  0h  e.  ~H
2221elimel 3751 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2321elimel 3751 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2422, 23normpythi 22597 . 2  |-  ( ( if ( A  e. 
~H ,  A ,  0h )  .ih  if ( B  e.  ~H ,  B ,  0h )
)  =  0  -> 
( ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) ) ) ^
2 )  =  ( ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 )  +  ( ( normh `  if ( B  e. 
~H ,  B ,  0h ) ) ^ 2 ) ) )
2510, 20, 24dedth2h 3741 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( (
normh `  B ) ^
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ifcif 3699   ` cfv 5413  (class class class)co 6040   0cc0 8946    + caddc 8949   2c2 10005   ^cexp 11337   ~Hchil 22375    +h cva 22376    .ih csp 22378   normhcno 22379   0hc0v 22380
This theorem is referenced by:  normpyc  22601  chscllem2  23093  hstnmoc  23679  hstpyth  23685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hfvadd 22456  ax-hv0cl 22459  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-hnorm 22424
  Copyright terms: Public domain W3C validator