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Theorem lgsdirnn0 23740
Description: Variation on lgsdir 23731 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 997 . . . . . 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 10908 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 23711 . . . . . . . . . . . . . 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 10991 . . . . . . . . . . . 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 9796 . . . . . . . . . 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 6312 . . . . . . . . . 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 2509 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
12 0z 10896 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 465 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 23734 . . . . . . . . . . . . . 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 14170 . . . . . . . . . . . . . . . . 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 14175 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2498 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2459 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 23739 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  /L 1 )  =  1 )
2322adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L 1 )  =  1 )
24 oveq2 6304 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  /L N )  =  ( x  /L 1 ) )
2524eqeq1d 2459 . . . . . . . . . . . . . . 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 6311 . . . . . . . . . 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 23711 . . . . . . . . . . . . 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 10991 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  CC )
3534mulid2d 9631 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
1  x.  ( 0  /L N ) )  =  ( 0  /L N ) )
3630, 35eqtr2d 2499 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
3711, 36pm2.61dane 2775 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) ) )
3837ralrimiva 2871 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
39383ad2ant3 1019 . . . . . 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 6303 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  /L N )  =  ( B  /L N ) )
4140oveq1d 6311 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( B  /L
N )  x.  (
0  /L N ) ) )
4241eqeq2d 2471 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) ) )
4342rspcv 3206 . . . . . 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 1019 . . . . . . . 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 10991 . . . . . 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 23711 . . . . . . . 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 10991 . . . . . 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 9634 . . . 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 2501 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( 0  /L N )  x.  ( B  /L
N ) ) )
56 oveq1 6303 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10890 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 1018 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9795 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2520 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 6311 . . 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 6311 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  /L N )  =  ( 0  /L
N ) )
6463oveq1d 6311 . . 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 2508 . 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 996 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 6303 . . . . . . . 8  |-  ( x  =  A  ->  (
x  /L N )  =  ( A  /L N ) )
6867oveq1d 6311 . . . . . . 7  |-  ( x  =  A  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( A  /L
N )  x.  (
0  /L N ) ) )
6968eqeq2d 2471 . . . . . 6  |-  ( x  =  A  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) ) )
7069rspcv 3206 . . . . 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 6304 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10991 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9796 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2520 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 6311 . . 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 6311 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  /L N )  =  ( 0  /L
N ) )
8079oveq2d 6312 . . 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 2508 . 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 23731 . . 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 1278 . 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 2776 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    x. cmul 9514   NN0cn0 10816   ZZcz 10885    gcd cgcd 14156    /Lclgs 23695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-phi 14308  df-pc 14373  df-lgs 23696
This theorem is referenced by:  lgsdchr  23749
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