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Theorem normpythi 21714
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 21689 . . 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 21680 . . . . . . . . 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 5838 . . . . . 6  |-  ( ( A  .ih  B )  =  0  ->  (
( A  .ih  B
)  +  ( B 
.ih  A ) )  =  ( 0  +  0 ) )
9 00id 8983 . . . . . 6  |-  ( 0  +  0 )  =  0
108, 9syl6eq 2333 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  (
( A  .ih  B
)  +  ( B 
.ih  A ) )  =  0 )
1110oveq2d 5836 . . . 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 21653 . . . . . 6  |-  ( A 
.ih  A )  e.  CC
132, 2hicli 21653 . . . . . 6  |-  ( B 
.ih  B )  e.  CC
1412, 13addcli 8837 . . . . 5  |-  ( ( A  .ih  A )  +  ( B  .ih  B ) )  e.  CC
1514addid1i 8995 . . . 4  |-  ( ( ( A  .ih  A
)  +  ( B 
.ih  B ) )  +  0 )  =  ( ( A  .ih  A )  +  ( B 
.ih  B ) )
1611, 15syl6eq 2333 . . 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 2329 . 2  |-  ( ( A  .ih  B )  =  0  ->  (
( A  +h  B
)  .ih  ( A  +h  B ) )  =  ( ( A  .ih  A )  +  ( B 
.ih  B ) ) )
181, 2hvaddcli 21591 . . 3  |-  ( A  +h  B )  e. 
~H
1918normsqi 21704 . 2  |-  ( (
normh `  ( A  +h  B ) ) ^
2 )  =  ( ( A  +h  B
)  .ih  ( A  +h  B ) )
201normsqi 21704 . . 3  |-  ( (
normh `  A ) ^
2 )  =  ( A  .ih  A )
212normsqi 21704 . . 3  |-  ( (
normh `  B ) ^
2 )  =  ( B  .ih  B )
2220, 21oveq12i 5832 . 2  |-  ( ( ( normh `  A ) ^ 2 )  +  ( ( normh `  B
) ^ 2 ) )  =  ( ( A  .ih  A )  +  ( B  .ih  B ) )
2317, 19, 223eqtr4g 2342 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 6    <-> wb 178    = wceq 1624    e. wcel 1685   ` cfv 5222  (class class class)co 5820   0cc0 8733    + caddc 8736   2c2 9791   ^cexp 11099   ~Hchil 21492    +h cva 21493    .ih csp 21495   normhcno 21496
This theorem is referenced by:  normpyth  21717  pjopythi  22291
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hfvadd 21573  ax-hv0cl 21576  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-hnorm 21541
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