MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipasslem11 Unicode version

Theorem ipasslem11 21418
Description: Lemma for ipassi 21419. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .s OLD `  U
)
ip1i.7  |-  P  =  ( .i OLD `  U
)
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem11.a  |-  A  e.  X
ipasslem11.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem11  |-  ( C  e.  CC  ->  (
( C S A ) P B )  =  ( C  x.  ( A P B ) ) )

Proof of Theorem ipasslem11
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 8834 . 2  |-  ( C  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  C  =  ( x  +  ( _i  x.  y ) ) )
2 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
3 recn 8827 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
4 mulcom 8823 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  =  ( y  x.  _i ) )
52, 3, 4sylancr 644 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  =  ( y  x.  _i ) )
65adantl 452 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  =  ( y  x.  _i ) )
76oveq2d 5874 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  =  ( x  +  ( y  x.  _i ) ) )
87eqeq2d 2294 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( _i  x.  y ) )  <-> 
C  =  ( x  +  ( y  x.  _i ) ) ) )
9 recn 8827 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
10 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
1110phnvi 21394 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 ipasslem11.a . . . . . . . . . 10  |-  A  e.  X
13 ip1i.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
14 ip1i.4 . . . . . . . . . . 11  |-  S  =  ( .s OLD `  U
)
1513, 14nvscl 21184 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  CC  /\  A  e.  X )  ->  (
x S A )  e.  X )
1611, 12, 15mp3an13 1268 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x S A )  e.  X )
179, 16syl 15 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x S A )  e.  X )
18 mulcl 8821 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  _i  e.  CC )  -> 
( y  x.  _i )  e.  CC )
193, 2, 18sylancl 643 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  x.  _i )  e.  CC )
2013, 14nvscl 21184 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y  x.  _i )  e.  CC  /\  A  e.  X )  ->  (
( y  x.  _i ) S A )  e.  X )
2111, 12, 20mp3an13 1268 . . . . . . . . 9  |-  ( ( y  x.  _i )  e.  CC  ->  (
( y  x.  _i ) S A )  e.  X )
2219, 21syl 15 . . . . . . . 8  |-  ( y  e.  RR  ->  (
( y  x.  _i ) S A )  e.  X )
23 ipasslem11.b . . . . . . . . 9  |-  B  e.  X
24 ip1i.2 . . . . . . . . . 10  |-  G  =  ( +v `  U
)
25 ip1i.7 . . . . . . . . . 10  |-  P  =  ( .i OLD `  U
)
2613, 24, 14, 25, 10ipdiri 21408 . . . . . . . . 9  |-  ( ( ( x S A )  e.  X  /\  ( ( y  x.  _i ) S A )  e.  X  /\  B  e.  X )  ->  ( ( ( x S A ) G ( ( y  x.  _i ) S A ) ) P B )  =  ( ( ( x S A ) P B )  +  ( ( ( y  x.  _i ) S A ) P B ) ) )
2723, 26mp3an3 1266 . . . . . . . 8  |-  ( ( ( x S A )  e.  X  /\  ( ( y  x.  _i ) S A )  e.  X )  ->  ( ( ( x S A ) G ( ( y  x.  _i ) S A ) ) P B )  =  ( ( ( x S A ) P B )  +  ( ( ( y  x.  _i ) S A ) P B ) ) )
2817, 22, 27syl2an 463 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x S A ) G ( ( y  x.  _i ) S A ) ) P B )  =  ( ( ( x S A ) P B )  +  ( ( ( y  x.  _i ) S A ) P B ) ) )
2913, 24, 14, 25, 10, 12, 23ipasslem9 21416 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( x S A ) P B )  =  ( x  x.  ( A P B ) ) )
3013, 14nvscl 21184 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  A  e.  X )  ->  (
_i S A )  e.  X )
3111, 2, 12, 30mp3an 1277 . . . . . . . . . 10  |-  ( _i S A )  e.  X
3213, 24, 14, 25, 10, 31, 23ipasslem9 21416 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y S ( _i S A ) ) P B )  =  ( y  x.  ( ( _i S A ) P B ) ) )
3313, 14nvsass 21186 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
y  e.  CC  /\  _i  e.  CC  /\  A  e.  X ) )  -> 
( ( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
3411, 33mpan 651 . . . . . . . . . . . 12  |-  ( ( y  e.  CC  /\  _i  e.  CC  /\  A  e.  X )  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
352, 12, 34mp3an23 1269 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
363, 35syl 15 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
3736oveq1d 5873 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y S ( _i S A ) ) P B ) )
3813, 25dipcl 21288 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
3911, 12, 23, 38mp3an 1277 . . . . . . . . . . . 12  |-  ( A P B )  e.  CC
40 mulass 8825 . . . . . . . . . . . 12  |-  ( ( y  e.  CC  /\  _i  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) ) )
412, 39, 40mp3an23 1269 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) ) )
423, 41syl 15 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) ) )
43 eqid 2283 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
4413, 24, 14, 25, 10, 12, 23, 43ipasslem10 21417 . . . . . . . . . . 11  |-  ( ( _i S A ) P B )  =  ( _i  x.  ( A P B ) )
4544oveq2i 5869 . . . . . . . . . 10  |-  ( y  x.  ( ( _i S A ) P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) )
4642, 45syl6eqr 2333 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
( _i S A ) P B ) ) )
4732, 37, 463eqtr4d 2325 . . . . . . . 8  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y  x.  _i )  x.  ( A P B ) ) )
4829, 47oveqan12d 5877 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x S A ) P B )  +  ( ( ( y  x.  _i ) S A ) P B ) )  =  ( ( x  x.  ( A P B ) )  +  ( ( y  x.  _i )  x.  ( A P B ) ) ) )
4928, 48eqtrd 2315 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x S A ) G ( ( y  x.  _i ) S A ) ) P B )  =  ( ( x  x.  ( A P B ) )  +  ( ( y  x.  _i )  x.  ( A P B ) ) ) )
5013, 24, 14nvdir 21189 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
x  e.  CC  /\  ( y  x.  _i )  e.  CC  /\  A  e.  X ) )  -> 
( ( x  +  ( y  x.  _i ) ) S A )  =  ( ( x S A ) G ( ( y  x.  _i ) S A ) ) )
5111, 50mpan 651 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  ( y  x.  _i )  e.  CC  /\  A  e.  X )  ->  (
( x  +  ( y  x.  _i ) ) S A )  =  ( ( x S A ) G ( ( y  x.  _i ) S A ) ) )
5212, 51mp3an3 1266 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  x.  _i )  e.  CC )  ->  ( ( x  +  ( y  x.  _i ) ) S A )  =  ( ( x S A ) G ( ( y  x.  _i ) S A ) ) )
539, 19, 52syl2an 463 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( y  x.  _i ) ) S A )  =  ( ( x S A ) G ( ( y  x.  _i ) S A ) ) )
5453oveq1d 5873 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x  +  ( y  x.  _i ) ) S A ) P B )  =  ( ( ( x S A ) G ( ( y  x.  _i ) S A ) ) P B ) )
55 adddir 8830 . . . . . . . 8  |-  ( ( x  e.  CC  /\  ( y  x.  _i )  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( x  +  ( y  x.  _i ) )  x.  ( A P B ) )  =  ( ( x  x.  ( A P B ) )  +  ( ( y  x.  _i )  x.  ( A P B ) ) ) )
5639, 55mp3an3 1266 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( y  x.  _i )  e.  CC )  ->  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) )  =  ( ( x  x.  ( A P B ) )  +  ( ( y  x.  _i )  x.  ( A P B ) ) ) )
579, 19, 56syl2an 463 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) )  =  ( ( x  x.  ( A P B ) )  +  ( ( y  x.  _i )  x.  ( A P B ) ) ) )
5849, 54, 573eqtr4d 2325 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( x  +  ( y  x.  _i ) ) S A ) P B )  =  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) ) )
59 oveq1 5865 . . . . . . 7  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C S A )  =  ( ( x  +  ( y  x.  _i ) ) S A ) )
6059oveq1d 5873 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  (
( C S A ) P B )  =  ( ( ( x  +  ( y  x.  _i ) ) S A ) P B ) )
61 oveq1 5865 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C  x.  ( A P B ) )  =  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) ) )
6260, 61eqeq12d 2297 . . . . 5  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  (
( ( C S A ) P B )  =  ( C  x.  ( A P B ) )  <->  ( (
( x  +  ( y  x.  _i ) ) S A ) P B )  =  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) ) ) )
6358, 62syl5ibrcom 213 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
648, 63sylbid 206 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( _i  x.  y ) )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
6564rexlimivv 2672 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  C  =  ( x  +  ( _i  x.  y
) )  ->  (
( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
661, 65syl 15 1  |-  ( C  e.  CC  ->  (
( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   _ici 8739    + caddc 8740    x. cmul 8742   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   normCVcnmcv 21146   .i OLDcdip 21273   CPreHil OLDccphlo 21390
This theorem is referenced by:  ipassi  21419
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  ax-addf 8816  ax-mulf 8817
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-iin 3908  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  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-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-cn 16957  df-cnp 16958  df-t1 17042  df-haus 17043  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ph 21391
  Copyright terms: Public domain W3C validator