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Theorem ipeq0 16542
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 2283 . . . . . 6  |-  ( * r `  F )  =  ( * r `
 F )
6 ip0l.z . . . . . 6  |-  Z  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16532 . . . . 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 972 . . . 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 956 . . . . 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 2618 . . . 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 15 . . 3  |-  ( W  e.  PreHil  ->  A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  ) )
12 oveq12 5867 . . . . . . 7  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
1312anidms 626 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
1413eqeq1d 2291 . . . . 5  |-  ( x  =  A  ->  (
( x  .,  x
)  =  Z  <->  ( A  .,  A )  =  Z ) )
15 eqeq1 2289 . . . . 5  |-  ( x  =  A  ->  (
x  =  .0.  <->  A  =  .0.  ) )
1614, 15imbi12d 311 . . . 4  |-  ( x  =  A  ->  (
( ( x  .,  x )  =  Z  ->  x  =  .0.  )  <->  ( ( A 
.,  A )  =  Z  ->  A  =  .0.  ) ) )
1716rspccva 2883 . . 3  |-  ( ( A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
1811, 17sylan 457 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
192, 3, 1, 6, 4ip0l 16540 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (  .0.  .,  A )  =  Z )
20 oveq1 5865 . . . 4  |-  ( A  =  .0.  ->  ( A  .,  A )  =  (  .0.  .,  A
) )
2120eqeq1d 2291 . . 3  |-  ( A  =  .0.  ->  (
( A  .,  A
)  =  Z  <->  (  .0.  .,  A )  =  Z ) )
2219, 21syl5ibrcom 213 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  ( A  =  .0.  ->  ( A  .,  A )  =  Z ) )
2318, 22impbid 183 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  <->  A  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   * rcstv 13210  Scalarcsca 13211   .icip 13213   0gc0g 13400   *Ringcsr 15609   LMHom clmhm 15776   LVecclvec 15855  ringLModcrglmod 15922   PreHilcphl 16528
This theorem is referenced by:  ip2eq  16557  ocvin  16574  lsmcss  16592  obsne0  16625  cphipeq0  18639  ipcau2  18664  tchcph  18667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530
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