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Theorem ipasslem11 25546
Description: Lemma for ipassi 25547. 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  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  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 9602 . 2  |-  ( C  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  C  =  ( x  +  ( _i  x.  y ) ) )
2 ax-icn 9561 . . . . . . . 8  |-  _i  e.  CC
3 recn 9592 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
4 mulcom 9588 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  =  ( y  x.  _i ) )
52, 3, 4sylancr 663 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  =  ( y  x.  _i ) )
65adantl 466 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  =  ( y  x.  _i ) )
76oveq2d 6310 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  =  ( x  +  ( y  x.  _i ) ) )
87eqeq2d 2481 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( _i  x.  y ) )  <-> 
C  =  ( x  +  ( y  x.  _i ) ) ) )
9 recn 9592 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
10 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
1110phnvi 25522 . . . . . . . . . 10  |-  U  e.  NrmCVec
12 ipasslem11.a . . . . . . . . . 10  |-  A  e.  X
13 ip1i.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
14 ip1i.4 . . . . . . . . . . 11  |-  S  =  ( .sOLD `  U )
1513, 14nvscl 25312 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  CC  /\  A  e.  X )  ->  (
x S A )  e.  X )
1611, 12, 15mp3an13 1315 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x S A )  e.  X )
179, 16syl 16 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x S A )  e.  X )
18 mulcl 9586 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  _i  e.  CC )  -> 
( y  x.  _i )  e.  CC )
193, 2, 18sylancl 662 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y  x.  _i )  e.  CC )
2013, 14nvscl 25312 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y  x.  _i )  e.  CC  /\  A  e.  X )  ->  (
( y  x.  _i ) S A )  e.  X )
2111, 12, 20mp3an13 1315 . . . . . . . . 9  |-  ( ( y  x.  _i )  e.  CC  ->  (
( y  x.  _i ) S A )  e.  X )
2219, 21syl 16 . . . . . . . 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  =  ( .iOLD `  U )
2613, 24, 14, 25, 10ipdiri 25536 . . . . . . . . 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 1313 . . . . . . . 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 477 . . . . . . 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 25544 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( x S A ) P B )  =  ( x  x.  ( A P B ) ) )
3013, 14nvscl 25312 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  A  e.  X )  ->  (
_i S A )  e.  X )
3111, 2, 12, 30mp3an 1324 . . . . . . . . . 10  |-  ( _i S A )  e.  X
3213, 24, 14, 25, 10, 31, 23ipasslem9 25544 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y S ( _i S A ) ) P B )  =  ( y  x.  ( ( _i S A ) P B ) ) )
3313, 14nvsass 25314 . . . . . . . . . . . . 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 670 . . . . . . . . . . . 12  |-  ( ( y  e.  CC  /\  _i  e.  CC  /\  A  e.  X )  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
352, 12, 34mp3an23 1316 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
363, 35syl 16 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( y  x.  _i ) S A )  =  ( y S ( _i S A ) ) )
3736oveq1d 6309 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y S ( _i S A ) ) P B ) )
3813, 25dipcl 25416 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
3911, 12, 23, 38mp3an 1324 . . . . . . . . . . . 12  |-  ( A P B )  e.  CC
40 mulass 9590 . . . . . . . . . . . 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 1316 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) ) )
423, 41syl 16 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) ) )
43 eqid 2467 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
4413, 24, 14, 25, 10, 12, 23, 43ipasslem10 25545 . . . . . . . . . . 11  |-  ( ( _i S A ) P B )  =  ( _i  x.  ( A P B ) )
4544oveq2i 6305 . . . . . . . . . 10  |-  ( y  x.  ( ( _i S A ) P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) )
4642, 45syl6eqr 2526 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
( _i S A ) P B ) ) )
4732, 37, 463eqtr4d 2518 . . . . . . . 8  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y  x.  _i )  x.  ( A P B ) ) )
4829, 47oveqan12d 6313 . . . . . . 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 2508 . . . . . 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 25317 . . . . . . . . . 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 670 . . . . . . . . 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 1313 . . . . . . . 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 477 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( y  x.  _i ) ) S A )  =  ( ( x S A ) G ( ( y  x.  _i ) S A ) ) )
5453oveq1d 6309 . . . . . 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 9597 . . . . . . . 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 1313 . . . . . . 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 477 . . . . . 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 2518 . . . . 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 6301 . . . . . . 7  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C S A )  =  ( ( x  +  ( y  x.  _i ) ) S A ) )
6059oveq1d 6309 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  (
( C S A ) P B )  =  ( ( ( x  +  ( y  x.  _i ) ) S A ) P B ) )
61 oveq1 6301 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C  x.  ( A P B ) )  =  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) ) )
6260, 61eqeq12d 2489 . . . . 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 222 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
648, 63sylbid 215 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( _i  x.  y ) )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
6564rexlimivv 2964 . 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 16 1  |-  ( C  e.  CC  ->  (
( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   ` cfv 5593  (class class class)co 6294   CCcc 9500   RRcr 9501   _ici 9504    + caddc 9505    x. cmul 9507   NrmCVeccnv 25268   +vcpv 25269   BaseSetcba 25270   .sOLDcns 25271   normCVcnmcv 25274   .iOLDcdip 25401   CPreHil OLDccphlo 25518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-fi 7881  df-sup 7911  df-oi 7945  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-icc 11546  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-sum 13484  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-hom 14591  df-cco 14592  df-rest 14690  df-topn 14691  df-0g 14709  df-gsum 14710  df-topgen 14711  df-pt 14712  df-prds 14715  df-xrs 14769  df-qtop 14774  df-imas 14775  df-xps 14777  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-mulg 15909  df-cntz 16204  df-cmn 16650  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-ntr 19366  df-cls 19367  df-cn 19573  df-cnp 19574  df-t1 19660  df-haus 19661  df-tx 19908  df-hmeo 20101  df-xms 20668  df-ms 20669  df-tms 20670  df-grpo 24984  df-gid 24985  df-ginv 24986  df-gdiv 24987  df-ablo 25075  df-vc 25230  df-nv 25276  df-va 25279  df-ba 25280  df-sm 25281  df-0v 25282  df-vs 25283  df-nmcv 25284  df-ims 25285  df-dip 25402  df-ph 25519
This theorem is referenced by:  ipassi  25547
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