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Theorem ipasslem5 24407
Description: Lemma for ipassi 24413. 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 11069 . . 3  |-  ( C  e.  QQ  <->  E. j  e.  ZZ  E. k  e.  NN  C  =  ( j  /  k ) )
2 zcn 10765 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
3 nnrecre 10472 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
43recnd 9526 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
5 ip1i.9 . . . . . . . . . . 11  |-  U  e.  CPreHil
OLD
65phnvi 24388 . . . . . . . . . 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 24282 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
116, 7, 10mp3an13 1306 . . . . . . . . 9  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
12 mulass 9484 . . . . . . . . 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 1261 . . . . . . . 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 10444 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  CC )
1615adantl 466 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  e.  CC )
17 nnne0 10468 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  =/=  0 )
1817adantl 466 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  k  =/=  0 )
1914, 16, 18divrecd 10224 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN )  ->  ( j  /  k
)  =  ( j  x.  ( 1  / 
k ) ) )
20193adant3 1008 . . . . . . . . 9  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
j  /  k )  =  ( j  x.  ( 1  /  k
) ) )
2120oveq1d 6218 . . . . . . . 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 6218 . . . . . . . . . . 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 24180 . . . . . . . . . . . . 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 1261 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  x.  (
1  /  k ) ) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2822, 27eqtrd 2495 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( j  /  k
) S A )  =  ( j S ( ( 1  / 
k ) S A ) ) )
2928oveq1d 6218 . . . . . . . . 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 24178 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
1  /  k )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  k
) S A )  e.  X )
316, 30mp3an1 1302 . . . . . . . . . . . 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 24405 . . . . . . . . . . 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 1184 . . . . . . . . 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 24406 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
38373adant1 1006 . . . . . . . . . 10  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( 1  / 
k ) S A ) P B )  =  ( ( 1  /  k )  x.  ( A P B ) ) )
3938oveq2d 6219 . . . . . . . . 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 2499 . . . . . . . 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 2506 . . . . . . 7  |-  ( ( j  e.  ZZ  /\  k  e.  NN  /\  A  e.  X )  ->  (
( ( j  / 
k ) S A ) P B )  =  ( ( j  /  k )  x.  ( A P B ) ) )
42 oveq1 6210 . . . . . . . . 9  |-  ( C  =  ( j  / 
k )  ->  ( C S A )  =  ( ( j  / 
k ) S A ) )
4342oveq1d 6218 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  (
( C S A ) P B )  =  ( ( ( j  /  k ) S A ) P B ) )
44 oveq1 6210 . . . . . . . 8  |-  ( C  =  ( j  / 
k )  ->  ( C  x.  ( A P B ) )  =  ( ( j  / 
k )  x.  ( A P B ) ) )
4543, 44eqeq12d 2476 . . . . . . 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 1190 . . . . 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 2952 . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    x. cmul 9401    / cdiv 10107   NNcn 10436   ZZcz 10760   QQcq 11067   NrmCVeccnv 24134   +vcpv 24135   BaseSetcba 24136   .sOLDcns 24137   .iOLDcdip 24267   CPreHil OLDccphlo 24384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-grpo 23850  df-gid 23851  df-ginv 23852  df-ablo 23941  df-vc 24096  df-nv 24142  df-va 24145  df-ba 24146  df-sm 24147  df-0v 24148  df-nmcv 24150  df-dip 24268  df-ph 24385
This theorem is referenced by:  ipasslem8  24409
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