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Theorem lgsdirnn0 21076
Description: Variation on lgsdir 21067 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 958 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  ZZ )
2 id 20 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  ZZ )
3 nn0z 10260 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 21047 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
52, 3, 4syl2anr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
65zcnd 10332 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  CC )
76adantr 452 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( x  / L N )  e.  CC )
87mul01d 9221 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( (
x  / L N
)  x.  0 )  =  0 )
9 simpr 448 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  0 )
109oveq2d 6056 . . . . . . . . . 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 2447 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
12 0z 10249 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 452 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 21070 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
1512, 13, 14sylancr 645 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
16 gcdcom 12975 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1712, 13, 16sylancr 645 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
18 nn0gcdid0 12980 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2436 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2412 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 21075 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  / L 1 )  =  1 )
2322adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L
1 )  =  1 )
24 oveq2 6048 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  / L N
)  =  ( x  / L 1 ) )
2524eqeq1d 2412 . . . . . . . . . . . . . . 15  |-  ( N  =  1  ->  (
( x  / L N )  =  1  <-> 
( x  / L
1 )  =  1 ) )
2623, 25syl5ibrcom 214 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  =  1  ->  ( x  / L N )  =  1 ) )
2721, 26sylbid 207 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  ->  ( x  / L N )  =  1 ) )
2815, 27sylbid 207 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  ->  ( x  / L N )  =  1 ) )
2928imp 419 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( x  / L N )  =  1 )
3029oveq1d 6055 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( (
x  / L N
)  x.  ( 0  / L N ) )  =  ( 1  x.  ( 0  / L N ) ) )
313ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  N  e.  ZZ )
32 lgscl 21047 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  / L N )  e.  ZZ )
3312, 31, 32sylancr 645 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  ZZ )
3433zcnd 10332 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  CC )
3534mulid2d 9062 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 1  x.  ( 0  / L N ) )  =  ( 0  / L N ) )
3630, 35eqtr2d 2437 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
3711, 36pm2.61dane 2645 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
3837ralrimiva 2749 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
39383ad2ant3 980 . . . . . 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 6047 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  / L N
)  =  ( B  / L N ) )
4140oveq1d 6055 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4241eqeq2d 2415 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( B  / L N )  x.  ( 0  / L N ) ) ) )
4342rspcv 3008 . . . . . 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 58 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( B  / L N
)  x.  ( 0  / L N ) ) )
4544adantr 452 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4633ad2ant3 980 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
4712, 46, 32sylancr 645 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  ZZ )
4847zcnd 10332 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  CC )
4948adantr 452 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  e.  CC )
50 lgscl 21047 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  / L N )  e.  ZZ )
511, 46, 50syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  ZZ )
5251zcnd 10332 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  CC )
5352adantr 452 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  / L N )  e.  CC )
5449, 53mulcomd 9065 . . . 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 2439 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( 0  / L N )  x.  ( B  / L N ) ) )
56 oveq1 6047 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10243 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 979 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9220 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2458 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 6055 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
62 simpr 448 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6362oveq1d 6055 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  / L N )  =  ( 0  / L N
) )
6463oveq1d 6055 . . 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 2446 . 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 957 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 6047 . . . . . . . 8  |-  ( x  =  A  ->  (
x  / L N
)  =  ( A  / L N ) )
6867oveq1d 6055 . . . . . . 7  |-  ( x  =  A  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
6968eqeq2d 2415 . . . . . 6  |-  ( x  =  A  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( A  / L N )  x.  ( 0  / L N ) ) ) )
7069rspcv 3008 . . . . 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 58 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( A  / L N
)  x.  ( 0  / L N ) ) )
7271adantr 452 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  / L N )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
73 oveq2 6048 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10332 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9221 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2458 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 6055 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
78 simpr 448 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
7978oveq1d 6055 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  / L N )  =  ( 0  / L N
) )
8079oveq2d 6056 . . 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 2446 . 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 21067 . . 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 1234 . 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 2646 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951   NN0cn0 10177   ZZcz 10238    gcd cgcd 12961    / Lclgs 21031
This theorem is referenced by:  lgsdchr  21085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110  df-pc 13166  df-lgs 21032
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