MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipasslem2 Structured version   Unicode version

Theorem ipasslem2 24383
Description: Lemma for ipassi 24392. Show the inner product associative law for nonpositive 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  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  U )
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem2  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( -u N  x.  ( A P B ) ) )

Proof of Theorem ipasslem2
StepHypRef Expression
1 nn0cn 10699 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
21negcld 9816 . . . 4  |-  ( N  e.  NN0  ->  -u N  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 24367 . . . . 5  |-  U  e.  NrmCVec
5 ipasslem1.b . . . . 5  |-  B  e.  X
6 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
7 ip1i.7 . . . . . 6  |-  P  =  ( .iOLD `  U )
86, 7dipcl 24261 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
94, 5, 8mp3an13 1306 . . . 4  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
10 mulcl 9476 . . . 4  |-  ( (
-u N  e.  CC  /\  ( A P B )  e.  CC )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
112, 9, 10syl2an 477 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
12 ip1i.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
136, 12nvscl 24157 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X )
144, 13mp3an1 1302 . . . . 5  |-  ( (
-u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
152, 14sylan 471 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
166, 7dipcl 24261 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u N S A )  e.  X  /\  B  e.  X )  ->  (
( -u N S A ) P B )  e.  CC )
174, 5, 16mp3an13 1306 . . . 4  |-  ( (
-u N S A )  e.  X  -> 
( ( -u N S A ) P B )  e.  CC )
1815, 17syl 16 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  e.  CC )
19 ax-1cn 9450 . . . . . . . . . . . . 13  |-  1  e.  CC
20 mulneg2 9892 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  -u 1
)  =  -u ( N  x.  1 ) )
2119, 20mpan2 671 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  ( N  x.  -u 1 )  =  -u ( N  x.  1 ) )
22 mulid1 9493 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  ( N  x.  1 )  =  N )
2322negeqd 9714 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  -u ( N  x.  1 )  =  -u N )
2421, 23eqtr2d 2496 . . . . . . . . . . 11  |-  ( N  e.  CC  ->  -u N  =  ( N  x.  -u 1 ) )
2524adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  A  e.  X )  -> 
-u N  =  ( N  x.  -u 1
) )
2625oveq1d 6214 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( ( N  x.  -u 1
) S A ) )
27 neg1cn 10535 . . . . . . . . . 10  |-  -u 1  e.  CC
286, 12nvsass 24159 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  ( N  e.  CC  /\  -u 1  e.  CC  /\  A  e.  X ) )  -> 
( ( N  x.  -u 1 ) S A )  =  ( N S ( -u 1 S A ) ) )
294, 28mpan 670 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( ( N  x.  -u 1 ) S A )  =  ( N S ( -u 1 S A ) ) )
3027, 29mp3an2 1303 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( ( N  x.  -u 1 ) S A )  =  ( N S ( -u 1 S A ) ) )
3126, 30eqtrd 2495 . . . . . . . 8  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
321, 31sylan 471 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
3332oveq1d 6214 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( ( N S ( -u
1 S A ) ) P B ) )
346, 12nvscl 24157 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
354, 27, 34mp3an12 1305 . . . . . . 7  |-  ( A  e.  X  ->  ( -u 1 S A )  e.  X )
36 ip1i.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
376, 36, 12, 7, 3, 5ipasslem1 24382 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( -u 1 S A )  e.  X )  ->  ( ( N S ( -u 1 S A ) ) P B )  =  ( N  x.  ( (
-u 1 S A ) P B ) ) )
3835, 37sylan2 474 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S ( -u 1 S A ) ) P B )  =  ( N  x.  ( (
-u 1 S A ) P B ) ) )
3933, 38eqtrd 2495 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( N  x.  ( ( -u
1 S A ) P B ) ) )
4039oveq2d 6215 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  -  (
( -u N S A ) P B ) )  =  ( (
-u N  x.  ( A P B ) )  -  ( N  x.  ( ( -u 1 S A ) P B ) ) ) )
416, 7dipcl 24261 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  (
( -u 1 S A ) P B )  e.  CC )
424, 5, 41mp3an13 1306 . . . . . . 7  |-  ( (
-u 1 S A )  e.  X  -> 
( ( -u 1 S A ) P B )  e.  CC )
4335, 42syl 16 . . . . . 6  |-  ( A  e.  X  ->  (
( -u 1 S A ) P B )  e.  CC )
44 mulcl 9476 . . . . . 6  |-  ( ( N  e.  CC  /\  ( ( -u 1 S A ) P B )  e.  CC )  ->  ( N  x.  ( ( -u 1 S A ) P B ) )  e.  CC )
451, 43, 44syl2an 477 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( N  x.  (
( -u 1 S A ) P B ) )  e.  CC )
4611, 45negsubd 9835 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  +  -u ( N  x.  (
( -u 1 S A ) P B ) ) )  =  ( ( -u N  x.  ( A P B ) )  -  ( N  x.  ( ( -u
1 S A ) P B ) ) ) )
47 mulneg1 9891 . . . . . . 7  |-  ( ( N  e.  CC  /\  ( ( -u 1 S A ) P B )  e.  CC )  ->  ( -u N  x.  ( ( -u 1 S A ) P B ) )  =  -u ( N  x.  (
( -u 1 S A ) P B ) ) )
481, 43, 47syl2an 477 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( ( -u 1 S A ) P B ) )  =  -u ( N  x.  (
( -u 1 S A ) P B ) ) )
4948oveq2d 6215 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  +  (
-u N  x.  (
( -u 1 S A ) P B ) ) )  =  ( ( -u N  x.  ( A P B ) )  +  -u ( N  x.  ( ( -u 1 S A ) P B ) ) ) )
502adantr 465 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  -> 
-u N  e.  CC )
519adantl 466 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( A P B )  e.  CC )
5243adantl 466 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u 1 S A ) P B )  e.  CC )
5350, 51, 52adddid 9520 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u 1 S A ) P B ) ) )  =  ( ( -u N  x.  ( A P B ) )  +  (
-u N  x.  (
( -u 1 S A ) P B ) ) ) )
546, 36, 12, 7, 3ipdiri 24381 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  ( ( A G ( -u 1 S A ) ) P B )  =  ( ( A P B )  +  ( (
-u 1 S A ) P B ) ) )
555, 54mp3an3 1304 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( -u 1 S A )  e.  X )  ->  ( ( A G ( -u 1 S A ) ) P B )  =  ( ( A P B )  +  ( (
-u 1 S A ) P B ) ) )
5635, 55mpdan 668 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )
57 eqid 2454 . . . . . . . . . . . . 13  |-  ( 0vec `  U )  =  (
0vec `  U )
586, 36, 12, 57nvrinv 24184 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
594, 58mpan 670 . . . . . . . . . . 11  |-  ( A  e.  X  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
6059oveq1d 6214 . . . . . . . . . 10  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( (
0vec `  U ) P B ) )
616, 57, 7dip0l 24267 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
624, 5, 61mp2an 672 . . . . . . . . . 10  |-  ( (
0vec `  U ) P B )  =  0
6360, 62syl6eq 2511 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  0 )
6456, 63eqtr3d 2497 . . . . . . . 8  |-  ( A  e.  X  ->  (
( A P B )  +  ( (
-u 1 S A ) P B ) )  =  0 )
6564oveq2d 6215 . . . . . . 7  |-  ( A  e.  X  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )  =  ( -u N  x.  0 ) )
662mul01d 9678 . . . . . . 7  |-  ( N  e.  NN0  ->  ( -u N  x.  0 )  =  0 )
6765, 66sylan9eqr 2517 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u 1 S A ) P B ) ) )  =  0 )
6853, 67eqtr3d 2497 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  +  (
-u N  x.  (
( -u 1 S A ) P B ) ) )  =  0 )
6949, 68eqtr3d 2497 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  +  -u ( N  x.  (
( -u 1 S A ) P B ) ) )  =  0 )
7040, 46, 693eqtr2d 2501 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  -  (
( -u N S A ) P B ) )  =  0 )
7111, 18, 70subeq0d 9837 . 2  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  =  ( (
-u N S A ) P B ) )
7271eqcomd 2462 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( -u N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5525  (class class class)co 6199   CCcc 9390   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397    - cmin 9705   -ucneg 9706   NN0cn0 10689   NrmCVeccnv 24113   +vcpv 24114   BaseSetcba 24115   .sOLDcns 24116   0veccn0v 24117   .iOLDcdip 24246   CPreHil OLDccphlo 24363
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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-grpo 23829  df-gid 23830  df-ginv 23831  df-ablo 23920  df-vc 24075  df-nv 24121  df-va 24124  df-ba 24125  df-sm 24126  df-0v 24127  df-nmcv 24129  df-dip 24247  df-ph 24364
This theorem is referenced by:  ipasslem3  24384
  Copyright terms: Public domain W3C validator