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Theorem pythi 21421
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 21415 . . 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 21387 . . . . . . . . 9  |-  U  e.  NrmCVec
101, 3diporthcom 21285 . . . . . . . . 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 188 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( B P A )  =  0 )
138, 12oveq12d 5838 . . . . . 6  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  ( 0  +  0 ) )
14 00id 8983 . . . . . 6  |-  ( 0  +  0 )  =  0
1513, 14syl6eq 2333 . . . . 5  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  0 )
1615oveq2d 5836 . . . 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 21281 . . . . . . 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 21281 . . . . . . 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 8837 . . . . 5  |-  ( ( A P A )  +  ( B P B ) )  e.  CC
2221addid1i 8995 . . . 4  |-  ( ( ( A P A )  +  ( B P B ) )  +  0 )  =  ( ( A P A )  +  ( B P B ) )
2316, 22syl6eq 2333 . . 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 2329 . 2  |-  ( ( A P B )  =  0  ->  (
( A G B ) P ( A G B ) )  =  ( ( A P A )  +  ( B P B ) ) )
251, 2nvgcl 21169 . . . 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 21279 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X )  ->  (
( A G B ) P ( A G B ) )  =  ( ( N `
 ( A G B ) ) ^
2 ) )
299, 26, 28mp2an 655 . 2  |-  ( ( A G B ) P ( A G B ) )  =  ( ( N `  ( A G B ) ) ^ 2 )
301, 27, 3ipidsq 21279 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )
319, 5, 30mp2an 655 . . 3  |-  ( A P A )  =  ( ( N `  A ) ^ 2 )
321, 27, 3ipidsq 21279 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B P B )  =  ( ( N `  B ) ^ 2 ) )
339, 6, 32mp2an 655 . . 3  |-  ( B P B )  =  ( ( N `  B ) ^ 2 )
3431, 33oveq12i 5832 . 2  |-  ( ( A P A )  +  ( B P B ) )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )
3524, 29, 343eqtr3g 2340 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 1624    e. wcel 1685   ` cfv 5222  (class class class)co 5820   CCcc 8731   0cc0 8733    + caddc 8736   2c2 9791   ^cexp 11099   NrmCVeccnv 21133   +vcpv 21134   BaseSetcba 21135   normCVcnmcv 21139   .i OLDcdip 21266   CPreHil OLDccphlo 21383
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-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  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-addf 8812  ax-mulf 8813
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-int 3865  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-se 4353  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-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  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-4 9802  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-fz 10778  df-fzo 10866  df-seq 11042  df-exp 11100  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-grpo 20851  df-gid 20852  df-ginv 20853  df-ablo 20942  df-vc 21095  df-nv 21141  df-va 21144  df-ba 21145  df-sm 21146  df-0v 21147  df-nmcv 21149  df-dip 21267  df-ph 21384
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