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Theorem ipeq0 16374
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
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 2253 . . . . . 6  |-  ( * r `  F )  =  ( * r `
 F )
6 ip0l.z . . . . . 6  |-  Z  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16364 . . . . 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 977 . . . 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 961 . . . . 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 2580 . . . 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 17 . . 3  |-  ( W  e.  PreHil  ->  A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  ) )
12 oveq12 5719 . . . . . . 7  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .,  x
)  =  ( A 
.,  A ) )
1312anidms 629 . . . . . 6  |-  ( x  =  A  ->  (
x  .,  x )  =  ( A  .,  A ) )
1413eqeq1d 2261 . . . . 5  |-  ( x  =  A  ->  (
( x  .,  x
)  =  Z  <->  ( A  .,  A )  =  Z ) )
15 eqeq1 2259 . . . . 5  |-  ( x  =  A  ->  (
x  =  .0.  <->  A  =  .0.  ) )
1614, 15imbi12d 313 . . . 4  |-  ( x  =  A  ->  (
( ( x  .,  x )  =  Z  ->  x  =  .0.  )  <->  ( ( A 
.,  A )  =  Z  ->  A  =  .0.  ) ) )
1716rcla4cva 2820 . . 3  |-  ( ( A. x  e.  V  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
1811, 17sylan 459 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  ->  A  =  .0.  )
)
192, 3, 1, 6, 4ip0l 16372 . . 3  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (  .0.  .,  A )  =  Z )
20 oveq1 5717 . . . 4  |-  ( A  =  .0.  ->  ( A  .,  A )  =  (  .0.  .,  A
) )
2120eqeq1d 2261 . . 3  |-  ( A  =  .0.  ->  (
( A  .,  A
)  =  Z  <->  (  .0.  .,  A )  =  Z ) )
2219, 21syl5ibrcom 215 . 2  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  ( A  =  .0.  ->  ( A  .,  A )  =  Z ) )
2318, 22impbid 185 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  (
( A  .,  A
)  =  Z  <->  A  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   Basecbs 13022   * rcstv 13084  Scalarcsca 13085   .icip 13087   0gc0g 13274   *Ringcsr 15444   LMHom clmhm 15611   LVecclvec 15690  ringLModcrglmod 15754   PreHilcphl 16360
This theorem is referenced by:  ip2eq  16389  ocvin  16406  lsmcss  16424  obsne0  16457  cphipeq0  18471  ipcau2  18496  tchcph  18499
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-plusg 13095  df-sca 13098  df-vsca 13099  df-0g 13278  df-mnd 14202  df-grp 14324  df-ghm 14516  df-lmod 15464  df-lmhm 15614  df-lvec 15691  df-sra 15757  df-rgmod 15758  df-phl 16362
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