MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipasslem5 Structured version   Unicode version

Theorem ipasslem5 25564
Description: Lemma for ipassi 25570. 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  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  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 11196 . . 3  |-  ( C  e.  QQ  <->  E. j  e.  ZZ  E. k  e.  NN  C  =  ( j  /  k ) )
2 zcn 10881 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
3 nnrecre 10584 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
43recnd 9634 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
5 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
65phnvi 25545 . . . . . . . . . 10  |-  U  e.  NrmCVec
7 ipasslem1.b . . . . . . . . . 10  |-  B  e.  X
8 ip1i.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
9 ip1i.7 . . . . . . . . . . 11  |-  P  =  ( .iOLD `  U )
108, 9dipcl 25439 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
116, 7, 10mp3an13 1315 . . . . . . . . 9  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
12 mulass 9592 . . . . . . . . 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 1270 . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  j  e.  CC )
15 nncn 10556 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  CC )
1615adantl 466 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  e.  CC )
17 nnne0 10580 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  =/=  0 )
1817adantl 466 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  =/=  0 )
1914, 16, 18divrecd 10335 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( j  /  k
)  =  ( j  x.  ( 1  / 
k ) ) )
20193adant3 1016 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
j  /  k )  =  ( j  x.  ( 1  /  k
) ) )
2120oveq1d 6310 . . . . . . . 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 6310 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( ( j  x.  ( 1  / 
k ) ) S A ) )
23 id 22 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  A  e.  X )
24 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .sOLD `  U )
258, 24nvsass 25337 . . . . . . . . . . . . 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 670 . . . . . . . . . . . 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 1270 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  x.  (
1  /  k ) ) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2822, 27eqtrd 2508 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2928oveq1d 6310 . . . . . . . . 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 25335 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  /  k )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  k
) S A )  e.  X )
316, 30mp3an1 1311 . . . . . . . . . . . 12  |-  ( ( ( 1  /  k
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  / 
k ) S A )  e.  X )
324, 31sylan 471 . . . . . . . . . . 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 25562 . . . . . . . . . . 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 474 . . . . . . . . . 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 1192 . . . . . . . . 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 25563 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
38373adant1 1014 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( 1  / 
k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
3938oveq2d 6311 . . . . . . . . 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 2512 . . . . . . . 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 2519 . . . . . . 7  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( ( j  /  k )  x.  ( A P B ) ) )
42 oveq1 6302 . . . . . . . . 9  |-  ( C  =  ( j  / 
k )  ->  ( C S A )  =  ( ( j  / 
k ) S A ) )
4342oveq1d 6310 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  (
( C S A ) P B )  =  ( ( ( j  /  k ) S A ) P B ) )
44 oveq1 6302 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  ( C  x.  ( A P B ) )  =  ( ( j  / 
k )  x.  ( A P B ) ) )
4543, 44eqeq12d 2489 . . . . . . 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 222 . . . . . 6  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  ( C  =  ( j  /  k )  -> 
( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
47463expia 1198 . . . . 5  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( A  e.  X  ->  ( C  =  ( j  /  k )  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4847com23 78 . . . 4  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( C  =  ( j  /  k )  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) ) )
4948rexlimivv 2964 . . 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 195 . 2  |-  ( C  e.  QQ  ->  ( A  e.  X  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) ) )
5150imp 429 1  |-  ( ( C  e.  QQ  /\  A  e.  X )  ->  ( ( 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    =/= wne 2662   E.wrex 2818   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    x. cmul 9509    / cdiv 10218   NNcn 10548   ZZcz 10876   QQcq 11194   NrmCVeccnv 25291   +vcpv 25292   BaseSetcba 25293   .sOLDcns 25294   .iOLDcdip 25424   CPreHil OLDccphlo 25541
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-grpo 25007  df-gid 25008  df-ginv 25009  df-ablo 25098  df-vc 25253  df-nv 25299  df-va 25302  df-ba 25303  df-sm 25304  df-0v 25305  df-nmcv 25307  df-dip 25425  df-ph 25542
This theorem is referenced by:  ipasslem8  25566
  Copyright terms: Public domain W3C validator