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Theorem ipeq0 16857
Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ip0l.z  |-  Z  =  ( 0g `  F
)
ip0l.o  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
ipeq0  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  <->  A  =  .0.  ) )

Proof of Theorem ipeq0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
2 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
3 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
4 ip0l.o . . . . . 6  |-  .0.  =  ( 0g `  W )
5 eqid 2435 . . . . . 6  |-  ( * r `  F )  =  ( * r `
 F )
6 ip0l.z . . . . . 6  |-  Z  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16847 . . . . 5  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (
( * r `  F ) `  (
x  .,  y )
)  =  ( y 
.,  x ) ) ) )
87simp3bi 974 . . . 4  |-  ( W  e.  PreHil  ->  A. x  e.  V  ( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  ( ( * r `
 F ) `  ( x  .,  y ) )  =  ( y 
.,  x ) ) )
9 simp2 958 . . . . 5  |-  ( ( ( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (
( * r `  F ) `  (
x  .,  y )
)  =  ( y 
.,  x ) )  ->  ( ( x 
.,  x )  =  Z  ->  x  =  .0.  ) )
109ralimi 2773 . . . 4  |-  ( A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (
( * r `  F ) `  (
x  .,  y )
)  =  ( y 
.,  x ) )  ->  A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  ) )
118, 10syl 16 . . 3  |-  ( W  e.  PreHil  ->  A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  ) )
12 oveq12 6081 . . . . . . 7  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
1312anidms 627 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
1413eqeq1d 2443 . . . . 5  |-  ( x  =  A  ->  (
( x  .,  x
)  =  Z  <->  ( A  .,  A )  =  Z ) )
15 eqeq1 2441 . . . . 5  |-  ( x  =  A  ->  (
x  =  .0.  <->  A  =  .0.  ) )
1614, 15imbi12d 312 . . . 4  |-  ( x  =  A  ->  (
( ( x  .,  x )  =  Z  ->  x  =  .0.  )  <->  ( ( A 
.,  A )  =  Z  ->  A  =  .0.  ) ) )
1716rspccva 3043 . . 3  |-  ( ( A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
1811, 17sylan 458 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
192, 3, 1, 6, 4ip0l 16855 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (  .0.  .,  A )  =  Z )
20 oveq1 6079 . . . 4  |-  ( A  =  .0.  ->  ( A  .,  A )  =  (  .0.  .,  A
) )
2120eqeq1d 2443 . . 3  |-  ( A  =  .0.  ->  (
( A  .,  A
)  =  Z  <->  (  .0.  .,  A )  =  Z ) )
2219, 21syl5ibrcom 214 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  ( A  =  .0.  ->  ( A  .,  A )  =  Z ) )
2318, 22impbid 184 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  <->  A  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   Basecbs 13457   * rcstv 13519  Scalarcsca 13520   .icip 13522   0gc0g 13711   *Ringcsr 15920   LMHom clmhm 16083   LVecclvec 16162  ringLModcrglmod 16229   PreHilcphl 16843
This theorem is referenced by:  ip2eq  16872  ocvin  16889  lsmcss  16907  obsne0  16940  cphipeq0  19154  ipcau2  19179  tchcph  19182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-plusg 13530  df-sca 13533  df-vsca 13534  df-0g 13715  df-mnd 14678  df-grp 14800  df-ghm 14992  df-lmod 15940  df-lmhm 16086  df-lvec 16163  df-sra 16232  df-rgmod 16233  df-phl 16845
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