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Theorem normpyth 22488
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 6020 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  .ih  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B
) )
21eqeq1d 2388 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  .ih  B
)  =  0  <->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  B )  =  0 ) )
3 oveq1 6020 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
43fveq2d 5665 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  ( A  +h  B ) )  =  ( normh `  ( if ( A  e.  ~H ,  A ,  0h )  +h  B ) ) )
54oveq1d 6028 . . . 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 5661 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( normh `  A )  =  ( normh `  if ( A  e.  ~H ,  A ,  0h ) ) )
76oveq1d 6028 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( normh `  A ) ^ 2 )  =  ( ( normh `  if ( A  e.  ~H ,  A ,  0h )
) ^ 2 ) )
87oveq1d 6028 . . . 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 2394 . . 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 6021 . . . 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 2388 . . 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 6021 . . . . . 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 5665 . . . . 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 6028 . . . 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 5661 . . . . . 6  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( normh `  B )  =  ( normh `  if ( B  e.  ~H ,  B ,  0h ) ) )
1716oveq1d 6028 . . . . 5  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( normh `  B ) ^ 2 )  =  ( ( normh `  if ( B  e.  ~H ,  B ,  0h )
) ^ 2 ) )
1817oveq2d 6029 . . . 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 2394 . . 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 22347 . . . 4  |-  0h  e.  ~H
2221elimel 3727 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
2321elimel 3727 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
2422, 23normpythi 22485 . 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 3717 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 1717   ifcif 3675   ` cfv 5387  (class class class)co 6013   0cc0 8916    + caddc 8919   2c2 9974   ^cexp 11302   ~Hchil 22263    +h cva 22264    .ih csp 22266   normhcno 22267   0hc0v 22268
This theorem is referenced by:  normpyc  22489  chscllem2  22981  hstnmoc  23567  hstpyth  23573
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-hfvadd 22344  ax-hv0cl 22347  ax-hvmul0 22354  ax-hfi 22422  ax-his1 22425  ax-his2 22426  ax-his3 22427  ax-his4 22428
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-hnorm 22312
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