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Theorem dip0r 24268
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  =  ( .iOLD `  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 24167 . . . 4  |-  ( U  e.  NrmCVec  ->  Z  e.  X
)
43adantr 465 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  Z  e.  X )
5 eqid 2454 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
6 eqid 2454 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
7 eqid 2454 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
8 dip0r.7 . . . 4  |-  P  =  ( .iOLD `  U )
91, 5, 6, 7, 8ipval2 24255 . . 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 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  /  4 ) )
104, 9mpd3an3 1316 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P Z )  =  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  /  4 ) )
11 neg1cn 10537 . . . . . . . . . . . . 13  |-  -u 1  e.  CC
126, 2nvsz 24171 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC )  ->  ( -u 1 ( .sOLD `  U ) Z )  =  Z )
1311, 12mpan2 671 . . . . . . . . . . . 12  |-  ( U  e.  NrmCVec  ->  ( -u 1
( .sOLD `  U ) Z )  =  Z )
1413adantr 465 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .sOLD `  U ) Z )  =  Z )
1514oveq2d 6217 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) )  =  ( A ( +v `  U
) Z ) )
1615fveq2d 5804 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) )  =  ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) )
1716oveq1d 6216 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .sOLD `  U ) Z ) ) ) ^ 2 )  =  ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) )
1817oveq2d 6217 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) ) )
191, 5, 6, 7, 8ipval2lem3 24253 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  RR )
204, 19mpd3an3 1316 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  RR )
2120recnd 9524 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) Z ) ) ^ 2 )  e.  CC )
2221subidd 9819 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 ) )  =  0 )
2318, 22eqtrd 2495 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  =  0 )
24 negicn 9723 . . . . . . . . . . . . . . 15  |-  -u _i  e.  CC
256, 2nvsz 24171 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  -u _i  e.  CC )  ->  ( -u _i ( .sOLD `  U ) Z )  =  Z )
2624, 25mpan2 671 . . . . . . . . . . . . . 14  |-  ( U  e.  NrmCVec  ->  ( -u _i ( .sOLD `  U
) Z )  =  Z )
27 ax-icn 9453 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
286, 2nvsz 24171 . . . . . . . . . . . . . . 15  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC )  ->  (
_i ( .sOLD `  U ) Z )  =  Z )
2927, 28mpan2 671 . . . . . . . . . . . . . 14  |-  ( U  e.  NrmCVec  ->  ( _i ( .sOLD `  U
) Z )  =  Z )
3026, 29eqtr4d 2498 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( -u _i ( .sOLD `  U
) Z )  =  ( _i ( .sOLD `  U ) Z ) )
3130adantr 465 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u _i ( .sOLD `  U ) Z )  =  ( _i ( .sOLD `  U
) Z ) )
3231oveq2d 6217 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) )  =  ( A ( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) )
3332fveq2d 5804 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) )  =  ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) )
3433oveq1d 6216 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) )
3534oveq2d 6217 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) )
361, 5, 6, 7, 8ipval2lem4 24254 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 )  e.  CC )
3727, 36mpan2 671 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  Z  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  e.  CC )
384, 37mpd3an3 1316 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  e.  CC )
3938subidd 9819 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  =  0 )
4035, 39eqtrd 2495 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .sOLD `  U ) Z ) ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  =  0 )
4140oveq2d 6217 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) )  =  ( _i  x.  0 ) )
4223, 41oveq12d 6219 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
43 it0e0 10659 . . . . . . 7  |-  ( _i  x.  0 )  =  0
4443oveq2i 6212 . . . . . 6  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
45 00id 9656 . . . . . 6  |-  ( 0  +  0 )  =  0
4644, 45eqtri 2483 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
4742, 46syl6eq 2511 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  =  0 )
4847oveq1d 6216 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  /  4 )  =  ( 0  /  4
) )
49 4cn 10511 . . . 4  |-  4  e.  CC
50 4ne0 10530 . . . 4  |-  4  =/=  0
5149, 50div0i 10177 . . 3  |-  ( 0  /  4 )  =  0
5248, 51syl6eq 2511 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) Z ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .sOLD `  U ) Z ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .sOLD `  U
) Z ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .sOLD `  U ) Z ) ) ) ^ 2 ) ) ) )  /  4 )  =  0 )
5310, 52eqtrd 2495 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A P Z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   CCcc 9392   RRcr 9393   0cc0 9394   1c1 9395   _ici 9396    + caddc 9397    x. cmul 9399    - cmin 9707   -ucneg 9708    / cdiv 10105   2c2 10483   4c4 10485   ^cexp 11983   NrmCVeccnv 24115   +vcpv 24116   BaseSetcba 24117   .sOLDcns 24118   0veccn0v 24119   normCVcnmcv 24121   .iOLDcdip 24248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-grpo 23831  df-gid 23832  df-ginv 23833  df-ablo 23922  df-vc 24077  df-nv 24123  df-va 24126  df-ba 24127  df-sm 24128  df-0v 24129  df-nmcv 24131  df-dip 24249
This theorem is referenced by:  dip0l  24269  siii  24406
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