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Theorem ipasslem5 21429
Description: Lemma for ipassi 21435. 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 10334 . . 3  |-  ( C  e.  QQ  <->  E. j  e.  ZZ  E. k  e.  NN  C  =  ( j  /  k ) )
2 zcn 10045 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
3 nnrecre 9798 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
43recnd 8877 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
5 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
65phnvi 21410 . . . . . . . . . 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 21304 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
116, 7, 10mp3an13 1268 . . . . . . . . 9  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
12 mulass 8841 . . . . . . . . 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 1224 . . . . . . . 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 451 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  j  e.  CC )
15 nncn 9770 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  CC )
1615adantl 452 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  e.  CC )
17 nnne0 9794 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  =/=  0 )
1817adantl 452 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  =/=  0 )
1914, 16, 18divrecd 9555 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( j  /  k
)  =  ( j  x.  ( 1  / 
k ) ) )
20193adant3 975 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
j  /  k )  =  ( j  x.  ( 1  /  k
) ) )
2120oveq1d 5889 . . . . . . . 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 5889 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( ( j  x.  ( 1  / 
k ) ) S A ) )
23 id 19 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  X )
24 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .s OLD `  U
)
258, 24nvsass 21202 . . . . . . . . . . . . 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 651 . . . . . . . . . . . 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 1224 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  x.  (
1  /  k ) ) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2822, 27eqtrd 2328 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2928oveq1d 5889 . . . . . . . . 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 21200 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  /  k )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  k
) S A )  e.  X )
316, 30mp3an1 1264 . . . . . . . . . . . 12  |-  ( ( ( 1  /  k
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  / 
k ) S A )  e.  X )
324, 31sylan 457 . . . . . . . . . . 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 21427 . . . . . . . . . . 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 460 . . . . . . . . . 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 1147 . . . . . . . . 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 21428 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
38373adant1 973 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( 1  / 
k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
3938oveq2d 5890 . . . . . . . . 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 2332 . . . . . . . 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 2339 . . . . . . 7  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( ( j  /  k )  x.  ( A P B ) ) )
42 oveq1 5881 . . . . . . . . 9  |-  ( C  =  ( j  / 
k )  ->  ( C S A )  =  ( ( j  / 
k ) S A ) )
4342oveq1d 5889 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  (
( C S A ) P B )  =  ( ( ( j  /  k ) S A ) P B ) )
44 oveq1 5881 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  ( C  x.  ( A P B ) )  =  ( ( j  / 
k )  x.  ( A P B ) ) )
4543, 44eqeq12d 2310 . . . . . . 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 213 . . . . . 6  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  ( C  =  ( j  /  k )  -> 
( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
47463expia 1153 . . . . 5  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( A  e.  X  ->  ( C  =  ( j  /  k )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4847com23 72 . . . 4  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( C  =  ( j  /  k )  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4948rexlimivv 2685 . . 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 187 . 2  |-  ( C  e.  QQ  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
5150imp 418 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   .i OLDcdip 21289   CPreHil OLDccphlo 21406
This theorem is referenced by:  ipasslem8  21431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-dip 21290  df-ph 21407
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