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Theorem pythi 24395
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. This is Metamath 100 proof #4. (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  =  ( .iOLD `  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  =  ( .iOLD `  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 24389 . . 3  |-  ( ( A G B ) P ( A G B ) )  =  ( ( ( A P A )  +  ( B P B ) )  +  ( ( A P B )  +  ( B P A ) ) )
8 id 22 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( A P B )  =  0 )
94phnvi 24361 . . . . . . . . 9  |-  U  e.  NrmCVec
101, 3diporthcom 24259 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A P B )  =  0  <->  ( B P A )  =  0 ) )
119, 5, 6, 10mp3an 1315 . . . . . . . 8  |-  ( ( A P B )  =  0  <->  ( B P A )  =  0 )
1211biimpi 194 . . . . . . 7  |-  ( ( A P B )  =  0  ->  ( B P A )  =  0 )
138, 12oveq12d 6211 . . . . . 6  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  ( 0  +  0 ) )
14 00id 9648 . . . . . 6  |-  ( 0  +  0 )  =  0
1513, 14syl6eq 2508 . . . . 5  |-  ( ( A P B )  =  0  ->  (
( A P B )  +  ( B P A ) )  =  0 )
1615oveq2d 6209 . . . 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 24255 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  ( A P A )  e.  CC )
189, 5, 5, 17mp3an 1315 . . . . . 6  |-  ( A P A )  e.  CC
191, 3dipcl 24255 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  B  e.  X )  ->  ( B P B )  e.  CC )
209, 6, 6, 19mp3an 1315 . . . . . 6  |-  ( B P B )  e.  CC
2118, 20addcli 9494 . . . . 5  |-  ( ( A P A )  +  ( B P B ) )  e.  CC
2221addid1i 9660 . . . 4  |-  ( ( ( A P A )  +  ( B P B ) )  +  0 )  =  ( ( A P A )  +  ( B P B ) )
2316, 22syl6eq 2508 . . 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 2504 . 2  |-  ( ( A P B )  =  0  ->  (
( A G B ) P ( A G B ) )  =  ( ( A P A )  +  ( B P B ) ) )
251, 2nvgcl 24143 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
269, 5, 6, 25mp3an 1315 . . 3  |-  ( A G B )  e.  X
27 pyth.6 . . . 4  |-  N  =  ( normCV `  U )
281, 27, 3ipidsq 24253 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X )  ->  (
( A G B ) P ( A G B ) )  =  ( ( N `
 ( A G B ) ) ^
2 ) )
299, 26, 28mp2an 672 . 2  |-  ( ( A G B ) P ( A G B ) )  =  ( ( N `  ( A G B ) ) ^ 2 )
301, 27, 3ipidsq 24253 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P A )  =  ( ( N `  A ) ^ 2 ) )
319, 5, 30mp2an 672 . . 3  |-  ( A P A )  =  ( ( N `  A ) ^ 2 )
321, 27, 3ipidsq 24253 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B P B )  =  ( ( N `  B ) ^ 2 ) )
339, 6, 32mp2an 672 . . 3  |-  ( B P B )  =  ( ( N `  B ) ^ 2 )
3431, 33oveq12i 6205 . 2  |-  ( ( A P A )  +  ( B P B ) )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )
3524, 29, 343eqtr3g 2515 1  |-  ( ( A P B )  =  0  ->  (
( N `  ( A G B ) ) ^ 2 )  =  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386    + caddc 9389   2c2 10475   ^cexp 11975   NrmCVeccnv 24107   +vcpv 24108   BaseSetcba 24109   normCVcnmcv 24113   .iOLDcdip 24240   CPreHil OLDccphlo 24357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-grpo 23823  df-gid 23824  df-ginv 23825  df-ablo 23914  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-nmcv 24123  df-dip 24241  df-ph 24358
This theorem is referenced by: (None)
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