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Theorem pythi 22304
Description: The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space  U. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
pyth.1  |-  X  =  ( BaseSet `  U )
pyth.2  |-  G  =  ( +v `  U
)
pyth.6  |-  N  =  ( normCV `  U )
pyth.7  |-  P  =  ( .i OLD `  U
)
pythi.u  |-  U  e.  CPreHil
OLD
pythi.a  |-  A  e.  X
pythi.b  |-  B  e.  X
Assertion
Ref Expression
pythi  |-  ( ( A P B )  =  0  ->  (
( N `  ( A G B ) ) ^ 2 )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) )

Proof of Theorem pythi
StepHypRef Expression
1 pyth.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 pyth.2 . . . 4  |-  G  =  ( +v `  U
)
3 pyth.7 . . . 4  |-  P  =  ( .i OLD `  U
)
4 pythi.u . . . 4  |-  U  e.  CPreHil
OLD
5 pythi.a . . . 4  |-  A  e.  X
6 pythi.b . . . 4  |-  B  e.  X
71, 2, 3, 4, 5, 6, 5, 6ip2dii 22298 . . 3  |-  ( ( A G B ) P ( A G B ) )  =  ( ( ( A P A )  +  ( B P B ) )  +  ( ( A P B )  +  ( B P A ) ) )
8 id 20 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( A P B )  =  0 )
94phnvi 22270 . . . . . . . . 9  |-  U  e.  NrmCVec
101, 3diporthcom 22168 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P B )  =  0  <->  ( B P A )  =  0 ) )
119, 5, 6, 10mp3an 1279 . . . . . . . 8  |-  ( ( A P B )  =  0  <->  ( B P A )  =  0 )
1211biimpi 187 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( B P A )  =  0 )
138, 12oveq12d 6058 . . . . . 6  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  ( 0  +  0 ) )
14 00id 9197 . . . . . 6  |-  ( 0  +  0 )  =  0
1513, 14syl6eq 2452 . . . . 5  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  0 )
1615oveq2d 6056 . . . 4  |-  ( ( A P B )  =  0  ->  (
( ( A P A )  +  ( B P B ) )  +  ( ( A P B )  +  ( B P A ) ) )  =  ( ( ( A P A )  +  ( B P B ) )  +  0 ) )
171, 3dipcl 22164 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  ( A P A )  e.  CC )
189, 5, 5, 17mp3an 1279 . . . . . 6  |-  ( A P A )  e.  CC
191, 3dipcl 22164 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  B  e.  X )  ->  ( B P B )  e.  CC )
209, 6, 6, 19mp3an 1279 . . . . . 6  |-  ( B P B )  e.  CC
2118, 20addcli 9050 . . . . 5  |-  ( ( A P A )  +  ( B P B ) )  e.  CC
2221addid1i 9209 . . . 4  |-  ( ( ( A P A )  +  ( B P B ) )  +  0 )  =  ( ( A P A )  +  ( B P B ) )
2316, 22syl6eq 2452 . . 3  |-  ( ( A P B )  =  0  ->  (
( ( A P A )  +  ( B P B ) )  +  ( ( A P B )  +  ( B P A ) ) )  =  ( ( A P A )  +  ( B P B ) ) )
247, 23syl5eq 2448 . 2  |-  ( ( A P B )  =  0  ->  (
( A G B ) P ( A G B ) )  =  ( ( A P A )  +  ( B P B ) ) )
251, 2nvgcl 22052 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
269, 5, 6, 25mp3an 1279 . . 3  |-  ( A G B )  e.  X
27 pyth.6 . . . 4  |-  N  =  ( normCV `  U )
281, 27, 3ipidsq 22162 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X )  ->  (
( A G B ) P ( A G B ) )  =  ( ( N `
 ( A G B ) ) ^
2 ) )
299, 26, 28mp2an 654 . 2  |-  ( ( A G B ) P ( A G B ) )  =  ( ( N `  ( A G B ) ) ^ 2 )
301, 27, 3ipidsq 22162 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )
319, 5, 30mp2an 654 . . 3  |-  ( A P A )  =  ( ( N `  A ) ^ 2 )
321, 27, 3ipidsq 22162 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B P B )  =  ( ( N `  B ) ^ 2 ) )
339, 6, 32mp2an 654 . . 3  |-  ( B P B )  =  ( ( N `  B ) ^ 2 )
3431, 33oveq12i 6052 . 2  |-  ( ( A P A )  +  ( B P B ) )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )
3524, 29, 343eqtr3g 2459 1  |-  ( ( A P B )  =  0  ->  (
( N `  ( A G B ) ) ^ 2 )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    + caddc 8949   2c2 10005   ^cexp 11337   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   normCVcnmcv 22022   .i OLDcdip 22149   CPreHil OLDccphlo 22266
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-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  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-addf 9025  ax-mulf 9026
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-int 4011  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-se 4502  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-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  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-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032  df-dip 22150  df-ph 22267
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