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

Theorem ipasslem2 22335
Description: Lemma for ipassi 22344. 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  =  ( .s OLD `  U
)
ip1i.7  |-  P  =  ( .i OLD `  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 10233 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
21negcld 9400 . . . 4  |-  ( N  e.  NN0  ->  -u N  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 22319 . . . . 5  |-  U  e.  NrmCVec
5 ipasslem1.b . . . . 5  |-  B  e.  X
6 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
7 ip1i.7 . . . . . 6  |-  P  =  ( .i OLD `  U
)
86, 7dipcl 22213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
94, 5, 8mp3an13 1271 . . . 4  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
10 mulcl 9076 . . . 4  |-  ( (
-u N  e.  CC  /\  ( A P B )  e.  CC )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
112, 9, 10syl2an 465 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
12 ip1i.4 . . . . . . 7  |-  S  =  ( .s OLD `  U
)
136, 12nvscl 22109 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X )
144, 13mp3an1 1267 . . . . 5  |-  ( (
-u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
152, 14sylan 459 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
166, 7dipcl 22213 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u N S A )  e.  X  /\  B  e.  X )  ->  (
( -u N S A ) P B )  e.  CC )
174, 5, 16mp3an13 1271 . . . 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 9050 . . . . . . . . . . . . 13  |-  1  e.  CC
20 mulneg2 9473 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  -u 1
)  =  -u ( N  x.  1 ) )
2119, 20mpan2 654 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  ( N  x.  -u 1 )  =  -u ( N  x.  1 ) )
22 mulid1 9090 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  ( N  x.  1 )  =  N )
2322negeqd 9302 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  -u ( N  x.  1 )  =  -u N )
2421, 23eqtr2d 2471 . . . . . . . . . . 11  |-  ( N  e.  CC  ->  -u N  =  ( N  x.  -u 1 ) )
2524adantr 453 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  A  e.  X )  -> 
-u N  =  ( N  x.  -u 1
) )
2625oveq1d 6098 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( ( N  x.  -u 1
) S A ) )
27 neg1cn 10069 . . . . . . . . . 10  |-  -u 1  e.  CC
286, 12nvsass 22111 . . . . . . . . . . 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 653 . . . . . . . . . 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 1268 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( ( N  x.  -u 1 ) S A )  =  ( N S ( -u 1 S A ) ) )
3126, 30eqtrd 2470 . . . . . . . 8  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
321, 31sylan 459 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
3332oveq1d 6098 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( ( N S ( -u
1 S A ) ) P B ) )
346, 12nvscl 22109 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
354, 27, 34mp3an12 1270 . . . . . . 7  |-  ( A  e.  X  ->  ( -u 1 S A )  e.  X )
36 ip1i.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
376, 36, 12, 7, 3, 5ipasslem1 22334 . . . . . . 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 462 . . . . . 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 2470 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( N  x.  ( ( -u
1 S A ) P B ) ) )
4039oveq2d 6099 . . . 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 22213 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  (
( -u 1 S A ) P B )  e.  CC )
424, 5, 41mp3an13 1271 . . . . . . 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 9076 . . . . . 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 465 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( N  x.  (
( -u 1 S A ) P B ) )  e.  CC )
4611, 45negsubd 9419 . . . 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 9472 . . . . . . 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 465 . . . . . 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 6099 . . . . 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 453 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  -> 
-u N  e.  CC )
519adantl 454 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( A P B )  e.  CC )
5243adantl 454 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u 1 S A ) P B )  e.  CC )
5350, 51, 52adddid 9114 . . . . . 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 22333 . . . . . . . . . . 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 1269 . . . . . . . . . 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 651 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )
57 eqid 2438 . . . . . . . . . . . . 13  |-  ( 0vec `  U )  =  (
0vec `  U )
586, 36, 12, 57nvrinv 22136 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
594, 58mpan 653 . . . . . . . . . . 11  |-  ( A  e.  X  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
6059oveq1d 6098 . . . . . . . . . 10  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( (
0vec `  U ) P B ) )
616, 57, 7dip0l 22219 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
624, 5, 61mp2an 655 . . . . . . . . . 10  |-  ( (
0vec `  U ) P B )  =  0
6360, 62syl6eq 2486 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  0 )
6456, 63eqtr3d 2472 . . . . . . . 8  |-  ( A  e.  X  ->  (
( A P B )  +  ( (
-u 1 S A ) P B ) )  =  0 )
6564oveq2d 6099 . . . . . . 7  |-  ( A  e.  X  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )  =  ( -u N  x.  0 ) )
662mul01d 9267 . . . . . . 7  |-  ( N  e.  NN0  ->  ( -u N  x.  0 )  =  0 )
6765, 66sylan9eqr 2492 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u 1 S A ) P B ) ) )  =  0 )
6853, 67eqtr3d 2472 . . . . 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 2472 . . . 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 2476 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  -  (
( -u N S A ) P B ) )  =  0 )
7111, 18, 70subeq0d 9421 . 2  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  =  ( (
-u N S A ) P B ) )
7271eqcomd 2443 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( -u N  x.  ( A P B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294   NN0cn0 10223   NrmCVeccnv 22065   +vcpv 22066   BaseSetcba 22067   .s
OLDcns 22068   0veccn0v 22069   .i
OLDcdip 22198   CPreHil OLDccphlo 22315
This theorem is referenced by:  ipasslem3  22336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-nmcv 22081  df-dip 22199  df-ph 22316
  Copyright terms: Public domain W3C validator