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Theorem pythi 21388
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 21382 . . 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 21354 . . . . . . . . 9  |-  U  e.  NrmCVec
101, 3diporthcom 21252 . . . . . . . . 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 5810 . . . . . 6  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  ( 0  +  0 ) )
14 00id 8955 . . . . . 6  |-  ( 0  +  0 )  =  0
1513, 14syl6eq 2306 . . . . 5  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  0 )
1615oveq2d 5808 . . . 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 21248 . . . . . . 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 21248 . . . . . . 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 8809 . . . . 5  |-  ( ( A P A )  +  ( B P B ) )  e.  CC
2221addid1i 8967 . . . 4  |-  ( ( ( A P A )  +  ( B P B ) )  +  0 )  =  ( ( A P A )  +  ( B P B ) )
2316, 22syl6eq 2306 . . 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 2302 . 2  |-  ( ( A P B )  =  0  ->  (
( A G B ) P ( A G B ) )  =  ( ( A P A )  +  ( B P B ) ) )
251, 2nvgcl 21136 . . . 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 21246 . . 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 21246 . . . 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 21246 . . . 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 5804 . 2  |-  ( ( A P A )  +  ( B P B ) )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )
3524, 29, 343eqtr3g 2313 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 4673  (class class class)co 5792   CCcc 8703   0cc0 8705    + caddc 8708   2c2 9763   ^cexp 11070   NrmCVeccnv 21100   +vcpv 21101   BaseSetcba 21102   normCVcnmcv 21106   .i OLDcdip 21233   CPreHil OLDccphlo 21350
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-grpo 20818  df-gid 20819  df-ginv 20820  df-ablo 20909  df-vc 21062  df-nv 21108  df-va 21111  df-ba 21112  df-sm 21113  df-0v 21114  df-nmcv 21116  df-dip 21234  df-ph 21351
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