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Theorem cncph 24364
Description: The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
cncph.6  |-  U  = 
<. <.  +  ,  x.  >. ,  abs >.
Assertion
Ref Expression
cncph  |-  U  e.  CPreHil
OLD

Proof of Theorem cncph
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cncph.6 . 2  |-  U  = 
<. <.  +  ,  x.  >. ,  abs >.
2 eqid 2451 . . . 4  |-  <. <.  +  ,  x.  >. ,  abs >.  = 
<. <.  +  ,  x.  >. ,  abs >.
32cnnv 24212 . . 3  |-  <. <.  +  ,  x.  >. ,  abs >.  e.  NrmCVec
4 mulm1 9890 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  ( -u 1  x.  y )  =  -u y )
54adantl 466 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( -u 1  x.  y )  =  -u y )
65oveq2d 6209 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  (
-u 1  x.  y
) )  =  ( x  +  -u y
) )
7 negsub 9761 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  -u y )  =  ( x  -  y ) )
86, 7eqtrd 2492 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  (
-u 1  x.  y
) )  =  ( x  -  y ) )
98fveq2d 5796 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( abs `  (
x  +  ( -u
1  x.  y ) ) )  =  ( abs `  ( x  -  y ) ) )
109oveq1d 6208 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  +  ( -u
1  x.  y ) ) ) ^ 2 )  =  ( ( abs `  ( x  -  y ) ) ^ 2 ) )
1110oveq2d 6209 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  +  ( -u 1  x.  y ) ) ) ^ 2 ) )  =  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  (
x  -  y ) ) ^ 2 ) ) )
12 sqabsadd 12882 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  +  y ) ) ^ 2 )  =  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  +  ( 2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) ) )
13 sqabssub 12883 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  -  y ) ) ^ 2 )  =  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) ) )
1412, 13oveq12d 6211 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  -  y ) ) ^ 2 ) )  =  ( ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  +  ( 2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) )  +  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) ) ) )
15 abscl 12878 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
1615recnd 9516 . . . . . . . . . 10  |-  ( x  e.  CC  ->  ( abs `  x )  e.  CC )
1716sqcld 12116 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( abs `  x
) ^ 2 )  e.  CC )
18 abscl 12878 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  ( abs `  y )  e.  RR )
1918recnd 9516 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( abs `  y )  e.  CC )
2019sqcld 12116 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
( abs `  y
) ^ 2 )  e.  CC )
21 addcl 9468 . . . . . . . . 9  |-  ( ( ( ( abs `  x
) ^ 2 )  e.  CC  /\  (
( abs `  y
) ^ 2 )  e.  CC )  -> 
( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  e.  CC )
2217, 20, 21syl2an 477 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  e.  CC )
23 2cn 10496 . . . . . . . . 9  |-  2  e.  CC
24 cjcl 12705 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
* `  y )  e.  CC )
25 mulcl 9470 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( * `  y
)  e.  CC )  ->  ( x  x.  ( * `  y
) )  e.  CC )
2624, 25sylan2 474 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  (
* `  y )
)  e.  CC )
27 recl 12710 . . . . . . . . . . 11  |-  ( ( x  x.  ( * `
 y ) )  e.  CC  ->  (
Re `  ( x  x.  ( * `  y
) ) )  e.  RR )
2827recnd 9516 . . . . . . . . . 10  |-  ( ( x  x.  ( * `
 y ) )  e.  CC  ->  (
Re `  ( x  x.  ( * `  y
) ) )  e.  CC )
2926, 28syl 16 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( Re `  (
x  x.  ( * `
 y ) ) )  e.  CC )
30 mulcl 9470 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( Re `  ( x  x.  ( * `  y ) ) )  e.  CC )  -> 
( 2  x.  (
Re `  ( x  x.  ( * `  y
) ) ) )  e.  CC )
3123, 29, 30sylancr 663 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( 2  x.  (
Re `  ( x  x.  ( * `  y
) ) ) )  e.  CC )
3222, 31, 22ppncand 9863 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  +  ( 2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) )  +  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( x  x.  ( * `  y
) ) ) ) ) )  =  ( ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  +  ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y
) ^ 2 ) ) ) )
3314, 32eqtrd 2492 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  -  y ) ) ^ 2 ) )  =  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  +  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
34 2times 10544 . . . . . . . 8  |-  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  e.  CC  ->  ( 2  x.  (
( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) )  =  ( ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y
) ^ 2 ) )  +  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
3534eqcomd 2459 . . . . . . 7  |-  ( ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) )  e.  CC  ->  ( ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y
) ^ 2 ) )  +  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) )  =  ( 2  x.  (
( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
3622, 35syl 16 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y
) ^ 2 ) )  +  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) )  =  ( 2  x.  (
( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
3733, 36eqtrd 2492 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  -  y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
3811, 37eqtrd 2492 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  +  ( -u 1  x.  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( abs `  x ) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) )
3938rgen2a 2893 . . 3  |-  A. x  e.  CC  A. y  e.  CC  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  (
x  +  ( -u
1  x.  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) )
40 addex 11093 . . . 4  |-  +  e.  _V
41 mulex 11094 . . . 4  |-  x.  e.  _V
42 absf 12936 . . . . 5  |-  abs : CC
--> RR
43 cnex 9467 . . . . 5  |-  CC  e.  _V
44 fex 6052 . . . . 5  |-  ( ( abs : CC --> RR  /\  CC  e.  _V )  ->  abs  e.  _V )
4542, 43, 44mp2an 672 . . . 4  |-  abs  e.  _V
46 cnaddablo 23982 . . . . . . 7  |-  +  e.  AbelOp
47 ablogrpo 23916 . . . . . . 7  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
4846, 47ax-mp 5 . . . . . 6  |-  +  e.  GrpOp
49 ax-addf 9465 . . . . . . 7  |-  +  :
( CC  X.  CC )
--> CC
5049fdmi 5665 . . . . . 6  |-  dom  +  =  ( CC  X.  CC )
5148, 50grporn 23844 . . . . 5  |-  CC  =  ran  +
5251isphg 24362 . . . 4  |-  ( (  +  e.  _V  /\  x.  e.  _V  /\  abs  e.  _V )  ->  ( <. <.  +  ,  x.  >. ,  abs >.  e.  CPreHil OLD  <->  (
<. <.  +  ,  x.  >. ,  abs >.  e.  NrmCVec  /\  A. x  e.  CC  A. y  e.  CC  (
( ( abs `  (
x  +  y ) ) ^ 2 )  +  ( ( abs `  ( x  +  (
-u 1  x.  y
) ) ) ^
2 ) )  =  ( 2  x.  (
( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) ) ) )
5340, 41, 45, 52mp3an 1315 . . 3  |-  ( <. <.  +  ,  x.  >. ,  abs >.  e.  CPreHil OLD  <->  ( <. <.  +  ,  x.  >. ,  abs >.  e.  NrmCVec  /\  A. x  e.  CC  A. y  e.  CC  ( ( ( abs `  ( x  +  y ) ) ^ 2 )  +  ( ( abs `  (
x  +  ( -u
1  x.  y ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( abs `  x
) ^ 2 )  +  ( ( abs `  y ) ^ 2 ) ) ) ) )
543, 39, 53mpbir2an 911 . 2  |-  <. <.  +  ,  x.  >. ,  abs >.  e.  CPreHil
OLD
551, 54eqeltri 2535 1  |-  U  e.  CPreHil
OLD
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071   <.cop 3984    X. cxp 4939   -->wf 5515   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   1c1 9387    + caddc 9389    x. cmul 9391    - cmin 9699   -ucneg 9700   2c2 10475   ^cexp 11975   *ccj 12696   Recre 12697   abscabs 12834   GrpOpcgr 23818   AbelOpcablo 23913   NrmCVeccnv 24107   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-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-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-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-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-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  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-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-grpo 23823  df-gid 23824  df-ablo 23914  df-vc 24069  df-nv 24115  df-ph 24358
This theorem is referenced by:  elimphu  24366  cnchl  24462
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