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Theorem lgsdirnn0 22637
Description: Variation on lgsdir 22628 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 984 . . . . . 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 10665 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 22608 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
52, 3, 4syl2anr 475 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
65zcnd 10744 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  CC )
76adantr 462 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
x  /L N )  e.  CC )
87mul01d 9564 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
( x  /L
N )  x.  0 )  =  0 )
9 simpr 458 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  0 )
109oveq2d 6106 . . . . . . . . . 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 2484 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
12 0z 10653 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 462 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 22631 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
1512, 13, 14sylancr 658 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
16 gcdcom 13700 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1712, 13, 16sylancr 658 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
18 nn0gcdid0 13705 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2473 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 22636 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  /L 1 )  =  1 )
2322adantl 463 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L 1 )  =  1 )
24 oveq2 6098 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  /L N )  =  ( x  /L 1 ) )
2524eqeq1d 2449 . . . . . . . . . . . . . . 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 6105 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( 1  x.  ( 0  /L N ) ) )
313ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  N  e.  ZZ )
32 lgscl 22608 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  /L
N )  e.  ZZ )
3312, 31, 32sylancr 658 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  ZZ )
3433zcnd 10744 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  CC )
3534mulid2d 9400 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
1  x.  ( 0  /L N ) )  =  ( 0  /L N ) )
3630, 35eqtr2d 2474 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
3711, 36pm2.61dane 2687 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) ) )
3837ralrimiva 2797 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
39383ad2ant3 1006 . . . . . 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 6097 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  /L N )  =  ( B  /L N ) )
4140oveq1d 6105 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( B  /L
N )  x.  (
0  /L N ) ) )
4241eqeq2d 2452 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) ) )
4342rspcv 3066 . . . . . 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 462 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L
N ) ) )
4633ad2ant3 1006 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
4712, 46, 32sylancr 658 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  ZZ )
4847zcnd 10744 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  CC )
4948adantr 462 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  e.  CC )
50 lgscl 22608 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  /L
N )  e.  ZZ )
511, 46, 50syl2anc 656 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  ZZ )
5251zcnd 10744 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  CC )
5352adantr 462 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  /L N )  e.  CC )
5449, 53mulcomd 9403 . . . 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 2476 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( 0  /L N )  x.  ( B  /L
N ) ) )
56 oveq1 6097 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10647 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 1005 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9563 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2495 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 6105 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
62 simpr 458 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6362oveq1d 6105 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  /L N )  =  ( 0  /L
N ) )
6463oveq1d 6105 . . 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 2483 . 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 983 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 6097 . . . . . . . 8  |-  ( x  =  A  ->  (
x  /L N )  =  ( A  /L N ) )
6867oveq1d 6105 . . . . . . 7  |-  ( x  =  A  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( A  /L
N )  x.  (
0  /L N ) ) )
6968eqeq2d 2452 . . . . . 6  |-  ( x  =  A  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) ) )
7069rspcv 3066 . . . . 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 462 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L
N ) ) )
73 oveq2 6098 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10744 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9564 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2495 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 6105 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
78 simpr 458 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
7978oveq1d 6105 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  /L N )  =  ( 0  /L
N ) )
8079oveq2d 6106 . . 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 2483 . 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 22628 . . 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 1263 . 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 2688 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283   NN0cn0 10575   ZZcz 10642    gcd cgcd 13686    /Lclgs 22592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-pc 13900  df-lgs 22593
This theorem is referenced by:  lgsdchr  22646
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