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Theorem lgsdirnn0 20594
Description: Variation on lgsdir 20585 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 10062 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  e.  ZZ )
4 lgscl 20565 . . . . . . . . . . . . . 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 10134 . . . . . . . . . . . 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 9027 . . . . . . . . . 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 5890 . . . . . . . . . 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 2339 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =  0 )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
12 0z 10051 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
133adantr 451 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
14 lgsne0 20588 . . . . . . . . . . . . . 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 12715 . . . . . . . . . . . . . . . . 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 12720 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
1918adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2017, 19eqtrd 2328 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2120eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
22 lgs1 20593 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  / L 1 )  =  1 )
2322adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  / L
1 )  =  1 )
24 oveq2 5882 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
x  / L N
)  =  ( x  / L 1 ) )
2524eqeq1d 2304 . . . . . . . . . . . . . . 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 5889 . . . . . . . . . 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 20565 . . . . . . . . . . . . 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 10134 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  e.  CC )
3534mulid2d 8869 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 1  x.  ( 0  / L N ) )  =  ( 0  / L N ) )
3630, 35eqtr2d 2329 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  / L N )  =/=  0
)  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  ( 0  / L N ) ) )
3711, 36pm2.61dane 2537 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) ) )
3837ralrimiva 2639 . . . . . . 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 5881 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  / L N
)  =  ( B  / L N ) )
4140oveq1d 5889 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( B  / L N )  x.  (
0  / L N
) ) )
4241eqeq2d 2307 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( B  / L N )  x.  ( 0  / L N ) ) ) )
4342rspcv 2893 . . . . . 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 10134 . . . . . 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 20565 . . . . . . . 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 10134 . . . . . 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 8872 . . . 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 2331 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  / L N )  =  ( ( 0  / L N )  x.  ( B  / L N ) ) )
56 oveq1 5881 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
57 zcn 10045 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
58573ad2ant2 977 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
5958mul02d 9026 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6056, 59sylan9eqr 2350 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6160oveq1d 5889 . . 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 5889 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  / L N )  =  ( 0  / L N
) )
6463oveq1d 5889 . . 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 2338 . 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 5881 . . . . . . . 8  |-  ( x  =  A  ->  (
x  / L N
)  =  ( A  / L N ) )
6867oveq1d 5889 . . . . . . 7  |-  ( x  =  A  ->  (
( x  / L N )  x.  (
0  / L N
) )  =  ( ( A  / L N )  x.  (
0  / L N
) ) )
6968eqeq2d 2307 . . . . . 6  |-  ( x  =  A  ->  (
( 0  / L N )  =  ( ( x  / L N )  x.  (
0  / L N
) )  <->  ( 0  / L N )  =  ( ( A  / L N )  x.  ( 0  / L N ) ) ) )
7069rspcv 2893 . . . . 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 5882 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7466zcnd 10134 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7574mul01d 9027 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7673, 75sylan9eqr 2350 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
7776oveq1d 5889 . . 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 5889 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  / L N )  =  ( 0  / L N
) )
8079oveq2d 5890 . . 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 2338 . 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 20585 . . 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 2538 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   NN0cn0 9981   ZZcz 10040    gcd cgcd 12701    / Lclgs 20549
This theorem is referenced by:  lgsdchr  20603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850  df-pc 12906  df-lgs 20550
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