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Theorem normpythi 22649
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.)
Hypotheses
Ref Expression
normsub.1  |-  A  e. 
~H
normsub.2  |-  B  e. 
~H
Assertion
Ref Expression
normpythi  |-  ( ( A  .ih  B )  =  0  ->  (
( normh `  ( A  +h  B ) ) ^
2 )  =  ( ( ( normh `  A
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) )

Proof of Theorem normpythi
StepHypRef Expression
1 normsub.1 . . . 4  |-  A  e. 
~H
2 normsub.2 . . . 4  |-  B  e. 
~H
31, 2, 1, 2normlem8 22624 . . 3  |-  ( ( A  +h  B ) 
.ih  ( A  +h  B ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  ( ( A  .ih  B
)  +  ( B 
.ih  A ) ) )
4 id 21 . . . . . . 7  |-  ( ( A  .ih  B )  =  0  ->  ( A  .ih  B )  =  0 )
5 orthcom 22615 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 ) )
61, 2, 5mp2an 655 . . . . . . . 8  |-  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 )
76biimpi 188 . . . . . . 7  |-  ( ( A  .ih  B )  =  0  ->  ( B  .ih  A )  =  0 )
84, 7oveq12d 6102 . . . . . 6  |-  ( ( A  .ih  B )  =  0  ->  (
( A  .ih  B
)  +  ( B 
.ih  A ) )  =  ( 0  +  0 ) )
9 00id 9246 . . . . . 6  |-  ( 0  +  0 )  =  0
108, 9syl6eq 2486 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  (
( A  .ih  B
)  +  ( B 
.ih  A ) )  =  0 )
1110oveq2d 6100 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  ( ( ( A 
.ih  A )  +  ( B  .ih  B
) )  +  0 ) )
121, 1hicli 22588 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
132, 2hicli 22588 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
1412, 13addcli 9099 . . . . 5  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
1514addid1i 9258 . . . 4  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  0 )  =  ( ( A  .ih  A )  +  ( B 
.ih  B ) )
1611, 15syl6eq 2486 . . 3  |-  ( ( A  .ih  B )  =  0  ->  (
( ( A  .ih  A )  +  ( B 
.ih  B ) )  +  ( ( A 
.ih  B )  +  ( B  .ih  A
) ) )  =  ( ( A  .ih  A )  +  ( B 
.ih  B ) ) )
173, 16syl5eq 2482 . 2  |-  ( ( A  .ih  B )  =  0  ->  (
( A  +h  B
)  .ih  ( A  +h  B ) )  =  ( ( A  .ih  A )  +  ( B 
.ih  B ) ) )
181, 2hvaddcli 22526 . . 3  |-  ( A  +h  B )  e. 
~H
1918normsqi 22639 . 2  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
201normsqi 22639 . . 3  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
212normsqi 22639 . . 3  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
2220, 21oveq12i 6096 . 2  |-  ( ( ( normh `  A ) ^ 2 )  +  ( ( normh `  B
) ^ 2 ) )  =  ( ( A  .ih  A )  +  ( B  .ih  B ) )
2317, 19, 223eqtr4g 2495 1  |-  ( ( 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    <-> wb 178    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   0cc0 8995    + caddc 8998   2c2 10054   ^cexp 11387   ~Hchil 22427    +h cva 22428    .ih csp 22430   normhcno 22431
This theorem is referenced by:  normpyth  22652  pjopythi  23226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-hfvadd 22508  ax-hv0cl 22511  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-hnorm 22476
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