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Theorem lgsdirnn0 20578
Description: Variation on lgsdir 20569 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 956 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  ZZ )
2 id 19 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  ->  x  e.  ZZ )
3 nn0z 10046 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 20549 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
52, 3, 4syl2anr 464 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  ZZ )
65zcnd 10118 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L N )  e.  CC )
76adantr 451 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( x  / L N )  e.  CC )
87mul01d 9011 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( (
x  / L N
)  x.  0 )  =  0 )
9 simpr 447 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  0 )
109oveq2d 5874 . . . . . . . . . 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 2326 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
12 0z 10035 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 451 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 20572 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
1512, 13, 14sylancr 644 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  <->  ( 0  gcd  N )  =  1 ) )
16 gcdcom 12699 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1712, 13, 16sylancr 644 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
18 nn0gcdid0 12704 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2315 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 20577 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  / L 1 )  =  1 )
2322adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L
1 )  =  1 )
24 oveq2 5866 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  / L N
)  =  ( x  / L 1 ) )
2524eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( N  =  1  ->  (
( x  / L N )  =  1  <-> 
( x  / L
1 )  =  1 ) )
2623, 25syl5ibrcom 213 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  =  1  ->  ( x  / L N )  =  1 ) )
2721, 26sylbid 206 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  ->  ( x  / L N )  =  1 ) )
2815, 27sylbid 206 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  / L N )  =/=  0  ->  ( x  / L N )  =  1 ) )
2928imp 418 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( x  / L N )  =  1 )
3029oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( (
x  / L N
)  x.  ( 0  / L N ) )  =  ( 1  x.  ( 0  / L N ) ) )
313ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  N  e.  ZZ )
32 lgscl 20549 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  / L N )  e.  ZZ )
3312, 31, 32sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  ZZ )
3433zcnd 10118 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  CC )
3534mulid2d 8853 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 1  x.  ( 0  / L N ) )  =  ( 0  / L N ) )
3630, 35eqtr2d 2316 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
3711, 36pm2.61dane 2524 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
3837ralrimiva 2626 . . . . . . 7  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
39383ad2ant3 978 . . . . . 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 5865 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  / L N
)  =  ( B  / L N ) )
4140oveq1d 5873 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4241eqeq2d 2294 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( B  / L N )  x.  ( 0  / L N ) ) ) )
4342rspcv 2880 . . . . . 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 56 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( B  / L N
)  x.  ( 0  / L N ) ) )
4544adantr 451 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4633ad2ant3 978 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
4712, 46, 32sylancr 644 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  ZZ )
4847zcnd 10118 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  e.  CC )
4948adantr 451 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  e.  CC )
50 lgscl 20549 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  / L N )  e.  ZZ )
511, 46, 50syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  ZZ )
5251zcnd 10118 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  / L N )  e.  CC )
5352adantr 451 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  / L N )  e.  CC )
5449, 53mulcomd 8856 . . . 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 2318 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( 0  / L N )  x.  ( B  / L N ) ) )
56 oveq1 5865 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10029 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 977 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9010 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2337 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 5873 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
62 simpr 447 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6362oveq1d 5873 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  / L N )  =  ( 0  / L N
) )
6463oveq1d 5873 . . 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 2325 . 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 955 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
67 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x  / L N
)  =  ( A  / L N ) )
6867oveq1d 5873 . . . . . . 7  |-  ( x  =  A  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
6968eqeq2d 2294 . . . . . 6  |-  ( x  =  A  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( A  / L N )  x.  ( 0  / L N ) ) ) )
7069rspcv 2880 . . . . 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 56 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  / L N
)  =  ( ( A  / L N
)  x.  ( 0  / L N ) ) )
7271adantr 451 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  / L N )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
73 oveq2 5866 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10118 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9011 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2337 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 5873 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  / L N )  =  ( 0  / L N
) )
78 simpr 447 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
7978oveq1d 5873 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  / L N )  =  ( 0  / L N
) )
8079oveq2d 5874 . . 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 2325 . 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 20569 . . 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 1232 . 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 2525 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   NN0cn0 9965   ZZcz 10024    gcd cgcd 12685    / Lclgs 20533
This theorem is referenced by:  lgsdchr  20587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-lgs 20534
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