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Theorem ipeq0 18433
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 2460 . . . . . 6  |-  ( *r `  F )  =  ( *r `  F )
6 ip0l.z . . . . . 6  |-  Z  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18423 . . . . 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 1008 . . . 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 992 . . . . 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 2850 . . . 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 6284 . . . . . . 7  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
1312anidms 645 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
1413eqeq1d 2462 . . . . 5  |-  ( x  =  A  ->  (
( x  .,  x
)  =  Z  <->  ( A  .,  A )  =  Z ) )
15 eqeq1 2464 . . . . 5  |-  ( x  =  A  ->  (
x  =  .0.  <->  A  =  .0.  ) )
1614, 15imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( ( x  .,  x )  =  Z  ->  x  =  .0.  )  <->  ( ( A 
.,  A )  =  Z  ->  A  =  .0.  ) ) )
1716rspccva 3206 . . 3  |-  ( ( A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
1811, 17sylan 471 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
192, 3, 1, 6, 4ip0l 18431 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (  .0.  .,  A )  =  Z )
20 oveq1 6282 . . . 4  |-  ( A  =  .0.  ->  ( A  .,  A )  =  (  .0.  .,  A
) )
2120eqeq1d 2462 . . 3  |-  ( A  =  .0.  ->  (
( A  .,  A
)  =  Z  <->  (  .0.  .,  A )  =  Z ) )
2219, 21syl5ibrcom 222 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  ( A  =  .0.  ->  ( A  .,  A )  =  Z ) )
2318, 22impbid 191 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  <->  A  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Basecbs 14479   *rcstv 14546  Scalarcsca 14547   .icip 14549   0gc0g 14684   *Ringcsr 17269   LMHom clmhm 17441   LVecclvec 17524  ringLModcrglmod 17591   PreHilcphl 18419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-sca 14560  df-vsca 14561  df-ip 14562  df-0g 14686  df-mnd 15721  df-grp 15851  df-ghm 16053  df-lmod 17290  df-lmhm 17444  df-lvec 17525  df-sra 17594  df-rgmod 17595  df-phl 18421
This theorem is referenced by:  ip2eq  18448  ocvin  18465  lsmcss  18483  obsne0  18516  cphipeq0  21378  ipcau2  21405  tchcph  21408
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