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Theorem lgsdirnn0 21124
Description: Variation on lgsdir 21115 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 959 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  ZZ )
2 id 21 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  ZZ )
3 nn0z 10305 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 21095 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
52, 3, 4syl2anr 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
65zcnd 10377 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  CC )
76adantr 453 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( x  / L N )  e.  CC )
87mul01d 9266 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( (
x  / L N
)  x.  0 )  =  0 )
9 simpr 449 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  0 )
109oveq2d 6098 . . . . . . . . . 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 2480 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
12 0z 10294 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 453 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 21118 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
1512, 13, 14sylancr 646 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
16 gcdcom 13021 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1712, 13, 16sylancr 646 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
18 nn0gcdid0 13026 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2469 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2445 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 21123 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  / L 1 )  =  1 )
2322adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L
1 )  =  1 )
24 oveq2 6090 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  / L N
)  =  ( x  / L 1 ) )
2524eqeq1d 2445 . . . . . . . . . . . . . . 15  |-  ( N  =  1  ->  (
( x  / L N )  =  1  <-> 
( x  / L
1 )  =  1 ) )
2623, 25syl5ibrcom 215 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  =  1  ->  ( x  / L N )  =  1 ) )
2721, 26sylbid 208 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  ->  ( x  / L N )  =  1 ) )
2815, 27sylbid 208 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  ->  ( x  / L N )  =  1 ) )
2928imp 420 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( x  / L N )  =  1 )
3029oveq1d 6097 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( (
x  / L N
)  x.  ( 0  / L N ) )  =  ( 1  x.  ( 0  / L N ) ) )
313ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  N  e.  ZZ )
32 lgscl 21095 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  / L N )  e.  ZZ )
3312, 31, 32sylancr 646 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  ZZ )
3433zcnd 10377 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  CC )
3534mulid2d 9107 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 1  x.  ( 0  / L N ) )  =  ( 0  / L N ) )
3630, 35eqtr2d 2470 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
3711, 36pm2.61dane 2683 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
3837ralrimiva 2790 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
39383ad2ant3 981 . . . . . 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 6089 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  / L N
)  =  ( B  / L N ) )
4140oveq1d 6097 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4241eqeq2d 2448 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( B  / L N )  x.  ( 0  / L N ) ) ) )
4342rspcv 3049 . . . . . 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 59 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( B  / L N
)  x.  ( 0  / L N ) ) )
4544adantr 453 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4633ad2ant3 981 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
4712, 46, 32sylancr 646 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  ZZ )
4847zcnd 10377 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  CC )
4948adantr 453 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  e.  CC )
50 lgscl 21095 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  / L N )  e.  ZZ )
511, 46, 50syl2anc 644 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  ZZ )
5251zcnd 10377 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  CC )
5352adantr 453 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  / L N )  e.  CC )
5449, 53mulcomd 9110 . . . 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 2472 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( 0  / L N )  x.  ( B  / L N ) ) )
56 oveq1 6089 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10288 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 980 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9265 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2491 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 6097 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
62 simpr 449 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6362oveq1d 6097 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  / L N )  =  ( 0  / L N
) )
6463oveq1d 6097 . . 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 2479 . 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 958 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 6089 . . . . . . . 8  |-  ( x  =  A  ->  (
x  / L N
)  =  ( A  / L N ) )
6867oveq1d 6097 . . . . . . 7  |-  ( x  =  A  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
6968eqeq2d 2448 . . . . . 6  |-  ( x  =  A  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( A  / L N )  x.  ( 0  / L N ) ) ) )
7069rspcv 3049 . . . . 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 59 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( A  / L N
)  x.  ( 0  / L N ) ) )
7271adantr 453 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  / L N )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
73 oveq2 6090 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10377 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9266 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2491 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 6097 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
78 simpr 449 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
7978oveq1d 6097 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  / L N )  =  ( 0  / L N
) )
8079oveq2d 6098 . . 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 2479 . 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 21115 . . 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 1235 . 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 2684 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706  (class class class)co 6082   CCcc 8989   0cc0 8991   1c1 8992    x. cmul 8996   NN0cn0 10222   ZZcz 10283    gcd cgcd 13007    / Lclgs 21079
This theorem is referenced by:  lgsdchr  21133
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-fz 11045  df-fzo 11137  df-fl 11203  df-mod 11252  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-dvds 12854  df-gcd 13008  df-prm 13081  df-phi 13156  df-pc 13212  df-lgs 21080
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