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Theorem ipasslem1 22332
Description: Lemma for ipassi 22342. Show the inner product associative law for nonnegative integers. (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
ipasslem1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem1
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0cn 10231 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  CC )
2 ax-1cn 9048 . . . . . . . . . . . 12  |-  1  e.  CC
3 ip1i.9 . . . . . . . . . . . . . 14  |-  U  e.  CPreHil
OLD
43phnvi 22317 . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
6 ip1i.2 . . . . . . . . . . . . . 14  |-  G  =  ( +v `  U
)
7 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .s OLD `  U
)
85, 6, 7nvdir 22112 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
k  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
94, 8mpan 652 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  X )  ->  (
( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
102, 9mp3an2 1267 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
111, 10sylan 458 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
125, 7nvsid 22108 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
134, 12mpan 652 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  (
1 S A )  =  A )
1413adantl 453 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1514oveq2d 6097 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k S A ) G ( 1 S A ) )  =  ( ( k S A ) G A ) )
1611, 15eqtrd 2468 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G A ) )
1716oveq1d 6096 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) G A ) P B ) )
18 ipasslem1.b . . . . . . . . . . . . 13  |-  B  e.  X
19 ip1i.7 . . . . . . . . . . . . . 14  |-  P  =  ( .i OLD `  U
)
205, 19dipcl 22211 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
214, 18, 20mp3an13 1270 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
2221mulid2d 9106 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2322adantl 453 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1  x.  ( A P B ) )  =  ( A P B ) )
2423oveq2d 6097 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
255, 7nvscl 22107 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  k  e.  CC  /\  A  e.  X )  ->  (
k S A )  e.  X )
264, 25mp3an1 1266 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( k S A )  e.  X )
271, 26sylan 458 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( k S A )  e.  X )
285, 6, 7, 19, 3ipdiri 22331 . . . . . . . . . . 11  |-  ( ( ( k S A )  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
2918, 28mp3an3 1268 . . . . . . . . . 10  |-  ( ( ( k S A )  e.  X  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3027, 29sylancom 649 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3124, 30eqtr4d 2471 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) G A ) P B ) )
3217, 31eqtr4d 2471 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) ) )
33 oveq1 6088 . . . . . . 7  |-  ( ( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  ->  (
( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3432, 33sylan9eq 2488 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
35 adddir 9083 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
362, 35mp3an2 1267 . . . . . . . 8  |-  ( ( k  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
371, 21, 36syl2an 464 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3837adantr 452 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3934, 38eqtr4d 2471 . . . . 5  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
4039exp31 588 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  X  ->  (
( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  -> 
( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
4140a2d 24 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  X  -> 
( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( A  e.  X  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
42 eqid 2436 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
435, 42, 19dip0l 22217 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
444, 18, 43mp2an 654 . . . 4  |-  ( (
0vec `  U ) P B )  =  0
455, 7, 42nv0 22118 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  ( 0vec `  U
) )
464, 45mpan 652 . . . . 5  |-  ( A  e.  X  ->  (
0 S A )  =  ( 0vec `  U
) )
4746oveq1d 6096 . . . 4  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( ( 0vec `  U ) P B ) )
4821mul02d 9264 . . . 4  |-  ( A  e.  X  ->  (
0  x.  ( A P B ) )  =  0 )
4944, 47, 483eqtr4a 2494 . . 3  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) )
50 oveq1 6088 . . . . . 6  |-  ( j  =  0  ->  (
j S A )  =  ( 0 S A ) )
5150oveq1d 6096 . . . . 5  |-  ( j  =  0  ->  (
( j S A ) P B )  =  ( ( 0 S A ) P B ) )
52 oveq1 6088 . . . . 5  |-  ( j  =  0  ->  (
j  x.  ( A P B ) )  =  ( 0  x.  ( A P B ) ) )
5351, 52eqeq12d 2450 . . . 4  |-  ( j  =  0  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) )
5453imbi2d 308 . . 3  |-  ( j  =  0  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) ) )
55 oveq1 6088 . . . . . 6  |-  ( j  =  k  ->  (
j S A )  =  ( k S A ) )
5655oveq1d 6096 . . . . 5  |-  ( j  =  k  ->  (
( j S A ) P B )  =  ( ( k S A ) P B ) )
57 oveq1 6088 . . . . 5  |-  ( j  =  k  ->  (
j  x.  ( A P B ) )  =  ( k  x.  ( A P B ) ) )
5856, 57eqeq12d 2450 . . . 4  |-  ( j  =  k  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
k S A ) P B )  =  ( k  x.  ( A P B ) ) ) )
5958imbi2d 308 . . 3  |-  ( j  =  k  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) ) ) )
60 oveq1 6088 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j S A )  =  ( ( k  +  1 ) S A ) )
6160oveq1d 6096 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j S A ) P B )  =  ( ( ( k  +  1 ) S A ) P B ) )
62 oveq1 6088 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  ( A P B ) )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
6361, 62eqeq12d 2450 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) )
6463imbi2d 308 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
65 oveq1 6088 . . . . . 6  |-  ( j  =  N  ->  (
j S A )  =  ( N S A ) )
6665oveq1d 6096 . . . . 5  |-  ( j  =  N  ->  (
( j S A ) P B )  =  ( ( N S A ) P B ) )
67 oveq1 6088 . . . . 5  |-  ( j  =  N  ->  (
j  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
6866, 67eqeq12d 2450 . . . 4  |-  ( j  =  N  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
6968imbi2d 308 . . 3  |-  ( j  =  N  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) ) )
7041, 49, 54, 59, 64, 69nn0indALT 10367 . 2  |-  ( N  e.  NN0  ->  ( A  e.  X  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
7170imp 419 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   0veccn0v 22067   .i
OLDcdip 22196   CPreHil OLDccphlo 22313
This theorem is referenced by:  ipasslem2  22333  ipasslem3  22334  ipasslem4  22335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079  df-dip 22197  df-ph 22314
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