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Theorem ipasslem5 22185
Description: Lemma for ipassi 22191. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-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
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem5  |-  ( ( C  e.  QQ  /\  A  e.  X )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )

Proof of Theorem ipasslem5
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 10509 . . 3  |-  ( C  e.  QQ  <->  E. j  e.  ZZ  E. k  e.  NN  C  =  ( j  /  k ) )
2 zcn 10220 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
3 nnrecre 9969 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
43recnd 9048 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
5 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
65phnvi 22166 . . . . . . . . . 10  |-  U  e.  NrmCVec
7 ipasslem1.b . . . . . . . . . 10  |-  B  e.  X
8 ip1i.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
9 ip1i.7 . . . . . . . . . . 11  |-  P  =  ( .i OLD `  U
)
108, 9dipcl 22060 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
116, 7, 10mp3an13 1270 . . . . . . . . 9  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
12 mulass 9012 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  ( 1  /  k
)  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( j  x.  ( 1  /  k
) )  x.  ( A P B ) )  =  ( j  x.  ( ( 1  / 
k )  x.  ( A P B ) ) ) )
132, 4, 11, 12syl3an 1226 . . . . . . . 8  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  x.  (
1  /  k ) )  x.  ( A P B ) )  =  ( j  x.  ( ( 1  / 
k )  x.  ( A P B ) ) ) )
142adantr 452 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  j  e.  CC )
15 nncn 9941 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  CC )
1615adantl 453 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  e.  CC )
17 nnne0 9965 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  =/=  0 )
1817adantl 453 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  =/=  0 )
1914, 16, 18divrecd 9726 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( j  /  k
)  =  ( j  x.  ( 1  / 
k ) ) )
20193adant3 977 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
j  /  k )  =  ( j  x.  ( 1  /  k
) ) )
2120oveq1d 6036 . . . . . . . 8  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
)  x.  ( A P B ) )  =  ( ( j  x.  ( 1  / 
k ) )  x.  ( A P B ) ) )
2220oveq1d 6036 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( ( j  x.  ( 1  / 
k ) ) S A ) )
23 id 20 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  X )
24 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .s OLD `  U
)
258, 24nvsass 21958 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
j  e.  CC  /\  ( 1  /  k
)  e.  CC  /\  A  e.  X )
)  ->  ( (
j  x.  ( 1  /  k ) ) S A )  =  ( j S ( ( 1  /  k
) S A ) ) )
266, 25mpan 652 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  ( 1  /  k
)  e.  CC  /\  A  e.  X )  ->  ( ( j  x.  ( 1  /  k
) ) S A )  =  ( j S ( ( 1  /  k ) S A ) ) )
272, 4, 23, 26syl3an 1226 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  x.  (
1  /  k ) ) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2822, 27eqtrd 2420 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2928oveq1d 6036 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( ( j S ( ( 1  /  k ) S A ) ) P B ) )
308, 24nvscl 21956 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  /  k )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  k
) S A )  e.  X )
316, 30mp3an1 1266 . . . . . . . . . . . 12  |-  ( ( ( 1  /  k
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  / 
k ) S A )  e.  X )
324, 31sylan 458 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  A  e.  X )  ->  ( ( 1  / 
k ) S A )  e.  X )
33 ip1i.2 . . . . . . . . . . . 12  |-  G  =  ( +v `  U
)
348, 33, 24, 9, 5, 7ipasslem3 22183 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  ( ( 1  / 
k ) S A )  e.  X )  ->  ( ( j S ( ( 1  /  k ) S A ) ) P B )  =  ( j  x.  ( ( ( 1  /  k
) S A ) P B ) ) )
3532, 34sylan2 461 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  ( k  e.  NN  /\  A  e.  X ) )  ->  ( (
j S ( ( 1  /  k ) S A ) ) P B )  =  ( j  x.  (
( ( 1  / 
k ) S A ) P B ) ) )
36353impb 1149 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j S ( ( 1  /  k
) S A ) ) P B )  =  ( j  x.  ( ( ( 1  /  k ) S A ) P B ) ) )
378, 33, 24, 9, 5, 7ipasslem4 22184 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
38373adant1 975 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( 1  / 
k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
3938oveq2d 6037 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
j  x.  ( ( ( 1  /  k
) S A ) P B ) )  =  ( j  x.  ( ( 1  / 
k )  x.  ( A P B ) ) ) )
4029, 36, 393eqtrd 2424 . . . . . . . 8  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( j  x.  ( ( 1  / 
k )  x.  ( A P B ) ) ) )
4113, 21, 403eqtr4rd 2431 . . . . . . 7  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( ( j  /  k )  x.  ( A P B ) ) )
42 oveq1 6028 . . . . . . . . 9  |-  ( C  =  ( j  / 
k )  ->  ( C S A )  =  ( ( j  / 
k ) S A ) )
4342oveq1d 6036 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  (
( C S A ) P B )  =  ( ( ( j  /  k ) S A ) P B ) )
44 oveq1 6028 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  ( C  x.  ( A P B ) )  =  ( ( j  / 
k )  x.  ( A P B ) ) )
4543, 44eqeq12d 2402 . . . . . . 7  |-  ( C  =  ( j  / 
k )  ->  (
( ( C S A ) P B )  =  ( C  x.  ( A P B ) )  <->  ( (
( j  /  k
) S A ) P B )  =  ( ( j  / 
k )  x.  ( A P B ) ) ) )
4641, 45syl5ibrcom 214 . . . . . 6  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  ( C  =  ( j  /  k )  -> 
( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
47463expia 1155 . . . . 5  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( A  e.  X  ->  ( C  =  ( j  /  k )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4847com23 74 . . . 4  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( C  =  ( j  /  k )  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4948rexlimivv 2779 . . 3  |-  ( E. j  e.  ZZ  E. k  e.  NN  C  =  ( j  / 
k )  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
501, 49sylbi 188 . 2  |-  ( C  e.  QQ  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
5150imp 419 1  |-  ( ( C  e.  QQ  /\  A  e.  X )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    x. cmul 8929    / cdiv 9610   NNcn 9933   ZZcz 10215   QQcq 10507   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s
OLDcns 21915   .i OLDcdip 22045   CPreHil OLDccphlo 22162
This theorem is referenced by:  ipasslem8  22187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fzo 11067  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-grpo 21628  df-gid 21629  df-ginv 21630  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928  df-dip 22046  df-ph 22163
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