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Theorem ipasslem2 21410
Description: Lemma for ipassi 21419. 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 9975 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
21negcld 9144 . . . 4  |-  ( N  e.  NN0  ->  -u N  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 21394 . . . . 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 21288 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
94, 5, 8mp3an13 1268 . . . 4  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
10 mulcl 8821 . . . 4  |-  ( (
-u N  e.  CC  /\  ( A P B )  e.  CC )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
112, 9, 10syl2an 463 . . 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 21184 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X )
144, 13mp3an1 1264 . . . . 5  |-  ( (
-u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
152, 14sylan 457 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
166, 7dipcl 21288 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u N S A )  e.  X  /\  B  e.  X )  ->  (
( -u N S A ) P B )  e.  CC )
174, 5, 16mp3an13 1268 . . . 4  |-  ( (
-u N S A )  e.  X  -> 
( ( -u N S A ) P B )  e.  CC )
1815, 17syl 15 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  e.  CC )
19 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
20 mulneg2 9217 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  -u 1
)  =  -u ( N  x.  1 ) )
2119, 20mpan2 652 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  ( N  x.  -u 1 )  =  -u ( N  x.  1 ) )
22 mulid1 8835 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  ( N  x.  1 )  =  N )
2322negeqd 9046 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  -u ( N  x.  1 )  =  -u N )
2421, 23eqtr2d 2316 . . . . . . . . . . 11  |-  ( N  e.  CC  ->  -u N  =  ( N  x.  -u 1 ) )
2524adantr 451 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  A  e.  X )  -> 
-u N  =  ( N  x.  -u 1
) )
2625oveq1d 5873 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( ( N  x.  -u 1
) S A ) )
27 neg1cn 9813 . . . . . . . . . 10  |-  -u 1  e.  CC
286, 12nvsass 21186 . . . . . . . . . . 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 651 . . . . . . . . . 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 1265 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( ( N  x.  -u 1 ) S A )  =  ( N S ( -u 1 S A ) ) )
3126, 30eqtrd 2315 . . . . . . . 8  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
321, 31sylan 457 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
3332oveq1d 5873 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( ( N S ( -u
1 S A ) ) P B ) )
346, 12nvscl 21184 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
354, 27, 34mp3an12 1267 . . . . . . 7  |-  ( A  e.  X  ->  ( -u 1 S A )  e.  X )
36 ip1i.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
376, 36, 12, 7, 3, 5ipasslem1 21409 . . . . . . 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 460 . . . . . 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 2315 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( N  x.  ( ( -u
1 S A ) P B ) ) )
4039oveq2d 5874 . . . 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 21288 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  (
( -u 1 S A ) P B )  e.  CC )
424, 5, 41mp3an13 1268 . . . . . . 7  |-  ( (
-u 1 S A )  e.  X  -> 
( ( -u 1 S A ) P B )  e.  CC )
4335, 42syl 15 . . . . . 6  |-  ( A  e.  X  ->  (
( -u 1 S A ) P B )  e.  CC )
44 mulcl 8821 . . . . . 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 463 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( N  x.  (
( -u 1 S A ) P B ) )  e.  CC )
4611, 45negsubd 9163 . . . 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 9216 . . . . . . 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 463 . . . . . 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 5874 . . . . 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 451 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  -> 
-u N  e.  CC )
519adantl 452 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( A P B )  e.  CC )
5243adantl 452 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u 1 S A ) P B )  e.  CC )
5350, 51, 52adddid 8859 . . . . . 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 21408 . . . . . . . . . . 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 1266 . . . . . . . . . 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 649 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )
57 eqid 2283 . . . . . . . . . . . . 13  |-  ( 0vec `  U )  =  (
0vec `  U )
586, 36, 12, 57nvrinv 21211 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
594, 58mpan 651 . . . . . . . . . . 11  |-  ( A  e.  X  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
6059oveq1d 5873 . . . . . . . . . 10  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( (
0vec `  U ) P B ) )
616, 57, 7dip0l 21294 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
624, 5, 61mp2an 653 . . . . . . . . . 10  |-  ( (
0vec `  U ) P B )  =  0
6360, 62syl6eq 2331 . . . . . . . . 9  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  0 )
6456, 63eqtr3d 2317 . . . . . . . 8  |-  ( A  e.  X  ->  (
( A P B )  +  ( (
-u 1 S A ) P B ) )  =  0 )
6564oveq2d 5874 . . . . . . 7  |-  ( A  e.  X  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )  =  ( -u N  x.  0 ) )
662mul01d 9011 . . . . . . 7  |-  ( N  e.  NN0  ->  ( -u N  x.  0 )  =  0 )
6765, 66sylan9eqr 2337 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u 1 S A ) P B ) ) )  =  0 )
6853, 67eqtr3d 2317 . . . . 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 2317 . . . 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 2321 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N  x.  ( A P B ) )  -  (
( -u N S A ) P B ) )  =  0 )
7111, 18, 70subeq0d 9165 . 2  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  =  ( (
-u N S A ) P B ) )
7271eqcomd 2288 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   NN0cn0 9965   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   .i
OLDcdip 21273   CPreHil OLDccphlo 21390
This theorem is referenced by:  ipasslem3  21411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-dip 21274  df-ph 21391
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