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Theorem pythi 21258
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 21252 . . 3  |-  ( ( A G B ) P ( A G B ) )  =  ( ( ( A P A )  +  ( B P B ) )  +  ( ( A P B )  +  ( B P A ) ) )
8 id 21 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( A P B )  =  0 )
94phnvi 21224 . . . . . . . . 9  |-  U  e.  NrmCVec
101, 3diporthcom 21122 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P B )  =  0  <->  ( B P A )  =  0 ) )
119, 5, 6, 10mp3an 1282 . . . . . . . 8  |-  ( ( A P B )  =  0  <->  ( B P A )  =  0 )
1211biimpi 188 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( B P A )  =  0 )
138, 12oveq12d 5728 . . . . . 6  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  ( 0  +  0 ) )
14 00id 8867 . . . . . 6  |-  ( 0  +  0 )  =  0
1513, 14syl6eq 2301 . . . . 5  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  0 )
1615oveq2d 5726 . . . 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 21118 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  ( A P A )  e.  CC )
189, 5, 5, 17mp3an 1282 . . . . . 6  |-  ( A P A )  e.  CC
191, 3dipcl 21118 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  B  e.  X )  ->  ( B P B )  e.  CC )
209, 6, 6, 19mp3an 1282 . . . . . 6  |-  ( B P B )  e.  CC
2118, 20addcli 8721 . . . . 5  |-  ( ( A P A )  +  ( B P B ) )  e.  CC
2221addid1i 8879 . . . 4  |-  ( ( ( A P A )  +  ( B P B ) )  +  0 )  =  ( ( A P A )  +  ( B P B ) )
2316, 22syl6eq 2301 . . 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 2297 . 2  |-  ( ( A P B )  =  0  ->  (
( A G B ) P ( A G B ) )  =  ( ( A P A )  +  ( B P B ) ) )
251, 2nvgcl 21006 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
269, 5, 6, 25mp3an 1282 . . 3  |-  ( A G B )  e.  X
27 pyth.6 . . . 4  |-  N  =  ( normCV `  U )
281, 27, 3ipidsq 21116 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X )  ->  (
( A G B ) P ( A G B ) )  =  ( ( N `
 ( A G B ) ) ^
2 ) )
299, 26, 28mp2an 656 . 2  |-  ( ( A G B ) P ( A G B ) )  =  ( ( N `  ( A G B ) ) ^ 2 )
301, 27, 3ipidsq 21116 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )
319, 5, 30mp2an 656 . . 3  |-  ( A P A )  =  ( ( N `  A ) ^ 2 )
321, 27, 3ipidsq 21116 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B P B )  =  ( ( N `  B ) ^ 2 ) )
339, 6, 32mp2an 656 . . 3  |-  ( B P B )  =  ( ( N `  B ) ^ 2 )
3431, 33oveq12i 5722 . 2  |-  ( ( A P A )  +  ( B P B ) )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )
3524, 29, 343eqtr3g 2308 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 6    <-> wb 178    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617    + caddc 8620   2c2 9675   ^cexp 10982   NrmCVeccnv 20970   +vcpv 20971   BaseSetcba 20972   normCVcnmcv 20976   .i OLDcdip 21103   CPreHil OLDccphlo 21220
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-grpo 20688  df-gid 20689  df-ginv 20690  df-ablo 20779  df-vc 20932  df-nv 20978  df-va 20981  df-ba 20982  df-sm 20983  df-0v 20984  df-nmcv 20986  df-dip 21104  df-ph 21221
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