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Theorem dip0r 21293
Description: Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
dip0r.1  |-  X  =  ( BaseSet `  U )
dip0r.5  |-  Z  =  ( 0vec `  U
)
dip0r.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dip0r  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P Z )  =  0 )

Proof of Theorem dip0r
StepHypRef Expression
1 dip0r.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 dip0r.5 . . . . 5  |-  Z  =  ( 0vec `  U
)
31, 2nvzcl 21192 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 451 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2283 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
6 eqid 2283 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 eqid 2283 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
8 dip0r.7 . . . 4  |-  P  =  ( .i OLD `  U
)
91, 5, 6, 7, 8ipval2 21280 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  ( A P Z )  =  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  / 
4 ) )
104, 9mpd3an3 1278 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P Z )  =  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  / 
4 ) )
11 neg1cn 9813 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
126, 2nvsz 21196 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC )  ->  ( -u 1 ( .s OLD `  U ) Z )  =  Z )
1311, 12mpan2 652 . . . . . . . . . . . 12  |-  ( U  e.  NrmCVec  ->  ( -u 1
( .s OLD `  U
) Z )  =  Z )
1413adantr 451 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) Z )  =  Z )
1514oveq2d 5874 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) )  =  ( A ( +v `  U
) Z ) )
1615fveq2d 5529 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) )  =  ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) )
1716oveq1d 5873 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) Z ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) )
1817oveq2d 5874 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) ) )
191, 5, 6, 7, 8ipval2lem3 21278 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  RR )
204, 19mpd3an3 1278 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  RR )
2120recnd 8861 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  CC )
2221subidd 9145 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) )  =  0 )
2318, 22eqtrd 2315 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  =  0 )
24 ax-icn 8796 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
2524negcli 9114 . . . . . . . . . . . . . . 15  |-  -u _i  e.  CC
266, 2nvsz 21196 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC )  ->  ( -u _i ( .s OLD `  U ) Z )  =  Z )
2725, 26mpan2 652 . . . . . . . . . . . . . 14  |-  ( U  e.  NrmCVec  ->  ( -u _i ( .s OLD `  U
) Z )  =  Z )
286, 2nvsz 21196 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC )  ->  (
_i ( .s OLD `  U ) Z )  =  Z )
2924, 28mpan2 652 . . . . . . . . . . . . . 14  |-  ( U  e.  NrmCVec  ->  ( _i ( .s OLD `  U
) Z )  =  Z )
3027, 29eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( -u _i ( .s OLD `  U
) Z )  =  ( _i ( .s
OLD `  U ) Z ) )
3130adantr 451 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i ( .s OLD `  U ) Z )  =  ( _i ( .s OLD `  U
) Z ) )
3231oveq2d 5874 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) )  =  ( A ( +v `  U ) ( _i ( .s
OLD `  U ) Z ) ) )
3332fveq2d 5529 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) )  =  ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) )
3433oveq1d 5873 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) )
3534oveq2d 5874 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) )
361, 5, 6, 7, 8ipval2lem4 21279 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  e.  CC )
3724, 36mpan2 652 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  e.  CC )
384, 37mpd3an3 1278 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  e.  CC )
3938subidd 9145 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) )  =  0 )
4035, 39eqtrd 2315 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) )  =  0 )
4140oveq2d 5874 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) )  =  ( _i  x.  0 ) )
4223, 41oveq12d 5876 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
4324mul01i 9002 . . . . . . 7  |-  ( _i  x.  0 )  =  0
4443oveq2i 5869 . . . . . 6  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
45 00id 8987 . . . . . 6  |-  ( 0  +  0 )  =  0
4644, 45eqtri 2303 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
4742, 46syl6eq 2331 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  =  0 )
4847oveq1d 5873 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  / 
4 )  =  ( 0  /  4 ) )
49 4cn 9820 . . . 4  |-  4  e.  CC
50 4re 9819 . . . . 5  |-  4  e.  RR
51 4pos 9832 . . . . 5  |-  0  <  4
5250, 51gt0ne0ii 9309 . . . 4  |-  4  =/=  0
5349, 52div0i 9494 . . 3  |-  ( 0  /  4 )  =  0
5448, 53syl6eq 2331 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) Z ) ) ) ^ 2 ) ) ) )  / 
4 )  =  0 )
5510, 54eqtrd 2315 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P Z )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   4c4 9797   ^cexp 11104   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   normCVcnmcv 21146   .i OLDcdip 21273
This theorem is referenced by:  dip0l  21294  siii  21431
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-inf2 7342  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  ax-pre-sup 8815
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-int 3863  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-dip 21274
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