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Theorem ipasslem11 24245
Description: Lemma for ipassi 24246. 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 9387 . 2  |-  ( C  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  C  =  ( x  +  ( _i  x.  y ) ) )
2 ax-icn 9346 . . . . . . . 8  |-  _i  e.  CC
3 recn 9377 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
4 mulcom 9373 . . . . . . . 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 6112 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  =  ( x  +  ( y  x.  _i ) ) )
87eqeq2d 2454 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( C  =  ( x  +  ( _i  x.  y ) )  <-> 
C  =  ( x  +  ( y  x.  _i ) ) ) )
9 recn 9377 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
10 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
1110phnvi 24221 . . . . . . . . . 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 24011 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  x  e.  CC  /\  A  e.  X )  ->  (
x S A )  e.  X )
1611, 12, 15mp3an13 1305 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x S A )  e.  X )
179, 16syl 16 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x S A )  e.  X )
18 mulcl 9371 . . . . . . . . . 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 24011 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
y  x.  _i )  e.  CC  /\  A  e.  X )  ->  (
( y  x.  _i ) S A )  e.  X )
2111, 12, 20mp3an13 1305 . . . . . . . . 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 24235 . . . . . . . . 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 1303 . . . . . . . 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 24243 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( x S A ) P B )  =  ( x  x.  ( A P B ) ) )
3013, 14nvscl 24011 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  _i  e.  CC  /\  A  e.  X )  ->  (
_i S A )  e.  X )
3111, 2, 12, 30mp3an 1314 . . . . . . . . . 10  |-  ( _i S A )  e.  X
3213, 24, 14, 25, 10, 31, 23ipasslem9 24243 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y S ( _i S A ) ) P B )  =  ( y  x.  ( ( _i S A ) P B ) ) )
3313, 14nvsass 24013 . . . . . . . . . . . . 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 1306 . . . . . . . . . . 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 6111 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y S ( _i S A ) ) P B ) )
3813, 25dipcl 24115 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
3911, 12, 23, 38mp3an 1314 . . . . . . . . . . . 12  |-  ( A P B )  e.  CC
40 mulass 9375 . . . . . . . . . . . 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 1306 . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( normCV `  U )  =  (
normCV
`  U )
4413, 24, 14, 25, 10, 12, 23, 43ipasslem10 24244 . . . . . . . . . . 11  |-  ( ( _i S A ) P B )  =  ( _i  x.  ( A P B ) )
4544oveq2i 6107 . . . . . . . . . 10  |-  ( y  x.  ( ( _i S A ) P B ) )  =  ( y  x.  (
_i  x.  ( A P B ) ) )
4642, 45syl6eqr 2493 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
( y  x.  _i )  x.  ( A P B ) )  =  ( y  x.  (
( _i S A ) P B ) ) )
4732, 37, 463eqtr4d 2485 . . . . . . . 8  |-  ( y  e.  RR  ->  (
( ( y  x.  _i ) S A ) P B )  =  ( ( y  x.  _i )  x.  ( A P B ) ) )
4829, 47oveqan12d 6115 . . . . . . 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 2475 . . . . . 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 24016 . . . . . . . . . 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 1303 . . . . . . . 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 6111 . . . . . 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 9382 . . . . . . . 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 1303 . . . . . . 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 2485 . . . . 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 6103 . . . . . . 7  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C S A )  =  ( ( x  +  ( y  x.  _i ) ) S A ) )
6059oveq1d 6111 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  (
( C S A ) P B )  =  ( ( ( x  +  ( y  x.  _i ) ) S A ) P B ) )
61 oveq1 6103 . . . . . 6  |-  ( C  =  ( x  +  ( y  x.  _i ) )  ->  ( C  x.  ( A P B ) )  =  ( ( x  +  ( y  x.  _i ) )  x.  ( A P B ) ) )
6260, 61eqeq12d 2457 . . . . 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 2851 . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2721   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   _ici 9289    + caddc 9290    x. cmul 9292   NrmCVeccnv 23967   +vcpv 23968   BaseSetcba 23969   .sOLDcns 23970   normCVcnmcv 23973   .iOLDcdip 24100   CPreHil OLDccphlo 24217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-cn 18836  df-cnp 18837  df-t1 18923  df-haus 18924  df-tx 19140  df-hmeo 19333  df-xms 19900  df-ms 19901  df-tms 19902  df-grpo 23683  df-gid 23684  df-ginv 23685  df-gdiv 23686  df-ablo 23774  df-vc 23929  df-nv 23975  df-va 23978  df-ba 23979  df-sm 23980  df-0v 23981  df-vs 23982  df-nmcv 23983  df-ims 23984  df-dip 24101  df-ph 24218
This theorem is referenced by:  ipassi  24246
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