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Theorem ipasslem2 25945
Description: Lemma for ipassi 25954. 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 10801 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
21negcld 9909 . . . 4  |-  ( N  e.  NN0  ->  -u N  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 25929 . . . . 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 25823 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
94, 5, 8mp3an13 1313 . . . 4  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
10 mulcl 9565 . . . 4  |-  ( (
-u N  e.  CC  /\  ( A P B )  e.  CC )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
112, 9, 10syl2an 475 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N  x.  ( A P B ) )  e.  CC )
12 ip1i.4 . . . . . . 7  |-  S  =  ( .sOLD `  U )
136, 12nvscl 25719 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X )
144, 13mp3an1 1309 . . . . 5  |-  ( (
-u N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
152, 14sylan 469 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  e.  X
)
166, 7dipcl 25823 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( -u N S A )  e.  X  /\  B  e.  X )  ->  (
( -u N S A ) P B )  e.  CC )
174, 5, 16mp3an13 1313 . . . 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 9539 . . . . . . . . . . . . 13  |-  1  e.  CC
20 mulneg2 9990 . . . . . . . . . . . . 13  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  -u 1
)  =  -u ( N  x.  1 ) )
2119, 20mpan2 669 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  ( N  x.  -u 1 )  =  -u ( N  x.  1 ) )
22 mulid1 9582 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  ( N  x.  1 )  =  N )
2322negeqd 9805 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  -u ( N  x.  1 )  =  -u N )
2421, 23eqtr2d 2496 . . . . . . . . . . 11  |-  ( N  e.  CC  ->  -u N  =  ( N  x.  -u 1 ) )
2524adantr 463 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  A  e.  X )  -> 
-u N  =  ( N  x.  -u 1
) )
2625oveq1d 6285 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  A  e.  X )  ->  ( -u N S A )  =  ( ( N  x.  -u 1
) S A ) )
27 neg1cn 10635 . . . . . . . . . 10  |-  -u 1  e.  CC
286, 12nvsass 25721 . . . . . . . . . . 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 668 . . . . . . . . . 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 1310 . . . . . . . . 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 469 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( -u N S A )  =  ( N S ( -u
1 S A ) ) )
3332oveq1d 6285 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u N S A ) P B )  =  ( ( N S ( -u
1 S A ) ) P B ) )
346, 12nvscl 25719 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 S A )  e.  X )
354, 27, 34mp3an12 1312 . . . . . . 7  |-  ( A  e.  X  ->  ( -u 1 S A )  e.  X )
36 ip1i.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
376, 36, 12, 7, 3, 5ipasslem1 25944 . . . . . . 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 472 . . . . . 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 6286 . . . 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 25823 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( -u 1 S A )  e.  X  /\  B  e.  X )  ->  (
( -u 1 S A ) P B )  e.  CC )
424, 5, 41mp3an13 1313 . . . . . . 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 9565 . . . . . 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 475 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( N  x.  (
( -u 1 S A ) P B ) )  e.  CC )
4611, 45negsubd 9928 . . . 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 9989 . . . . . . 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 475 . . . . . 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 6286 . . . . 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 463 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  -> 
-u N  e.  CC )
519adantl 464 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( A P B )  e.  CC )
5243adantl 464 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( -u 1 S A ) P B )  e.  CC )
5350, 51, 52adddid 9609 . . . . . 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 25943 . . . . . . . . . . 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 1311 . . . . . . . . . 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 666 . . . . . . . . 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 25746 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
594, 58mpan 668 . . . . . . . . . . 11  |-  ( A  e.  X  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
6059oveq1d 6285 . . . . . . . . . 10  |-  ( A  e.  X  ->  (
( A G (
-u 1 S A ) ) P B )  =  ( (
0vec `  U ) P B ) )
616, 57, 7dip0l 25829 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
624, 5, 61mp2an 670 . . . . . . . . . 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 6286 . . . . . . 7  |-  ( A  e.  X  ->  ( -u N  x.  ( ( A P B )  +  ( ( -u
1 S A ) P B ) ) )  =  ( -u N  x.  0 ) )
662mul01d 9768 . . . . . . 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 9930 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797   NN0cn0 10791   NrmCVeccnv 25675   +vcpv 25676   BaseSetcba 25677   .sOLDcns 25678   0veccn0v 25679   .iOLDcdip 25808   CPreHil OLDccphlo 25925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-grpo 25391  df-gid 25392  df-ginv 25393  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-nmcv 25691  df-dip 25809  df-ph 25926
This theorem is referenced by:  ipasslem3  25946
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