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Theorem lgsdirnn0 22678
Description: Variation on lgsdir 22669 valid for all  A ,  B but only for positive  N. (The exact location of the failure of this law is for  A  =  0,  B  <  0,  N  =  -u 1 in which case  ( 0  /L -u 1
)  =  1 but  ( B  /L -u 1 )  = 
-u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
Assertion
Ref Expression
lgsdirnn0  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )

Proof of Theorem lgsdirnn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 989 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  ZZ )
2 id 22 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  ZZ )
3 nn0z 10669 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 22649 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
52, 3, 4syl2anr 478 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
65zcnd 10748 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  CC )
76adantr 465 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
x  /L N )  e.  CC )
87mul01d 9568 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
( x  /L
N )  x.  0 )  =  0 )
9 simpr 461 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  0 )
109oveq2d 6107 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( x  /L
N )  x.  0 ) )
118, 10, 93eqtr4rd 2486 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
12 0z 10657 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 465 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 22672 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
1512, 13, 14sylancr 663 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
16 gcdcom 13704 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1712, 13, 16sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
18 nn0gcdid0 13709 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2475 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 22677 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  /L 1 )  =  1 )
2322adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L 1 )  =  1 )
24 oveq2 6099 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  /L N )  =  ( x  /L 1 ) )
2524eqeq1d 2451 . . . . . . . . . . . . . . 15  |-  ( N  =  1  ->  (
( x  /L
N )  =  1  <-> 
( x  /L 1 )  =  1 ) )
2623, 25syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  =  1  ->  ( x  /L N )  =  1 ) )
2721, 26sylbid 215 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  ->  ( x  /L N )  =  1 ) )
2815, 27sylbid 215 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  ->  ( x  /L N )  =  1 ) )
2928imp 429 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
x  /L N )  =  1 )
3029oveq1d 6106 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( 1  x.  ( 0  /L N ) ) )
313ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  N  e.  ZZ )
32 lgscl 22649 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  /L
N )  e.  ZZ )
3312, 31, 32sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  ZZ )
3433zcnd 10748 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  CC )
3534mulid2d 9404 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
1  x.  ( 0  /L N ) )  =  ( 0  /L N ) )
3630, 35eqtr2d 2476 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
3711, 36pm2.61dane 2689 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) ) )
3837ralrimiva 2799 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
39383ad2ant3 1011 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
40 oveq1 6098 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  /L N )  =  ( B  /L N ) )
4140oveq1d 6106 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( B  /L
N )  x.  (
0  /L N ) ) )
4241eqeq2d 2454 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) ) )
4342rspcv 3069 . . . . . 6  |-  ( B  e.  ZZ  ->  ( A. x  e.  ZZ  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  ->  (
0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) ) )
441, 39, 43sylc 60 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) )
4544adantr 465 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L
N ) ) )
4633ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
4712, 46, 32sylancr 663 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  ZZ )
4847zcnd 10748 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  CC )
4948adantr 465 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  e.  CC )
50 lgscl 22649 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  /L
N )  e.  ZZ )
511, 46, 50syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  ZZ )
5251zcnd 10748 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  CC )
5352adantr 465 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  /L N )  e.  CC )
5449, 53mulcomd 9407 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( 0  /L N )  x.  ( B  /L N ) )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) )
5545, 54eqtr4d 2478 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( 0  /L N )  x.  ( B  /L
N ) ) )
56 oveq1 6098 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10651 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 1010 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9567 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2497 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 6106 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
62 simpr 461 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6362oveq1d 6106 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  /L N )  =  ( 0  /L
N ) )
6463oveq1d 6106 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  /L N )  x.  ( B  /L N ) )  =  ( ( 0  /L N )  x.  ( B  /L N ) ) )
6555, 61, 643eqtr4d 2485 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( ( A  /L N )  x.  ( B  /L
N ) ) )
66 simp1 988 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 6098 . . . . . . . 8  |-  ( x  =  A  ->  (
x  /L N )  =  ( A  /L N ) )
6867oveq1d 6106 . . . . . . 7  |-  ( x  =  A  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( A  /L
N )  x.  (
0  /L N ) ) )
6968eqeq2d 2454 . . . . . 6  |-  ( x  =  A  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) ) )
7069rspcv 3069 . . . . 5  |-  ( A  e.  ZZ  ->  ( A. x  e.  ZZ  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  ->  (
0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) ) )
7166, 39, 70sylc 60 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) )
7271adantr 465 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L
N ) ) )
73 oveq2 6099 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10748 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9568 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2497 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 6106 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
78 simpr 461 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
7978oveq1d 6106 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  /L N )  =  ( 0  /L
N ) )
8079oveq2d 6107 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  /L N )  x.  ( B  /L N ) )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) )
8172, 77, 803eqtr4d 2485 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( ( A  /L N )  x.  ( B  /L
N ) ) )
82 lgsdir 22669 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
833, 82syl3anl3 1268 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
8465, 81, 83pm2.61da2ne 2690 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    x. cmul 9287   NN0cn0 10579   ZZcz 10646    gcd cgcd 13690    /Lclgs 22633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-phi 13841  df-pc 13904  df-lgs 22634
This theorem is referenced by:  lgsdchr  22687
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