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Theorem lgsdinn0 20595
Description: Variation on lgsdi 20587 valid for all  M ,  N but only for positive  A. (The exact location of the failure of this law is for  A  =  -u
1,  M  =  0, and some  N in which case  ( -u 1  / L 0 )  =  1 but  ( -u 1  / L N )  = 
-u 1 when  -u 1 is not a quadratic residue mod  N.) (Contributed by Mario Carneiro, 28-Apr-2016.)
Assertion
Ref Expression
lgsdinn0  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )

Proof of Theorem lgsdinn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2 sq1 11214 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2306 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 9990 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 10007 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 8853 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 9313 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 11192 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  0  <_  1 ) )  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  =  1
) )
96, 7, 8mpanr12 666 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
104, 5, 9syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  = 
1 ) )
1110adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
123, 11syl5bbr 250 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  1  <-> 
A  =  1 ) )
1312biimpa 470 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  = 
1 )
1413oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  ( 1  / L
x ) )
15 1lgs 20592 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
1  / L x )  =  1 )
1615ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  / L x )  =  1 )
1714, 16eqtrd 2328 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  1 )
1817oveq1d 5889 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  / L x )  x.  ( A  / L 0 ) )  =  ( 1  x.  ( A  / L
0 ) ) )
19 nn0z 10062 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10051 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 20565 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A  / L
0 )  e.  ZZ )
2320, 21, 22sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  e.  ZZ )
2423zcnd 10134 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  e.  CC )
2524mulid2d 8869 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  / L 0 ) )  =  ( A  / L 0 ) )
2618, 25eqtr2d 2329 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
27 lgscl 20565 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  ZZ )
2819, 27sylan 457 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  ZZ )
2928zcnd 10134 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  CC )
3029adantr 451 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L x )  e.  CC )
3130mul01d 9027 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( ( A  / L x )  x.  0 )  =  0 )
3219adantr 451 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  A  e.  ZZ )
33 lgs0 20564 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A  / L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
3432, 33syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
0 )  =  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) )
35 ifnefalse 3586 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2348 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  0 )
3736oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( ( A  / L x )  x.  ( A  / L 0 ) )  =  ( ( A  / L x )  x.  0 ) )
3831, 37, 363eqtr4rd 2339 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
3926, 38pm2.61dane 2537 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) ) )
4039ralrimiva 2639 . . . . . . 7  |-  ( A  e.  NN0  ->  A. x  e.  ZZ  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
41403ad2ant1 976 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  A. x  e.  ZZ  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
42 oveq2 5882 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  / L x )  =  ( A  / L N ) )
4342oveq1d 5889 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L N
)  x.  ( A  / L 0 ) ) )
4443eqeq2d 2307 . . . . . . 7  |-  ( x  =  N  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) ) )
4544rspcv 2893 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A. x  e.  ZZ  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L 0 ) )  ->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) ) )
461, 41, 45sylc 56 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) )
4746adantr 451 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) )
48193ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  A  e.  ZZ )
4948, 21, 22sylancl 643 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  e.  ZZ )
5049zcnd 10134 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  e.  CC )
5150adantr 451 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  e.  CC )
52 lgscl 20565 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
5348, 1, 52syl2anc 642 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
5453zcnd 10134 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  CC )
5554adantr 451 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L N )  e.  CC )
5651, 55mulcomd 8872 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( A  / L 0 )  x.  ( A  / L N ) )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) )
5747, 56eqtr4d 2331 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L 0 )  x.  ( A  / L N ) ) )
58 oveq1 5881 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 10134 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9026 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2350 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 5890 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( A  / L
0 ) )
63 simpr 447 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
6463oveq2d 5890 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L M )  =  ( A  / L
0 ) )
6564oveq1d 5889 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( A  / L M )  x.  ( A  / L N ) )  =  ( ( A  / L 0 )  x.  ( A  / L N ) ) )
6657, 62, 653eqtr4d 2338 . 2  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
67 simp2 956 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
68 oveq2 5882 . . . . . . . 8  |-  ( x  =  M  ->  ( A  / L x )  =  ( A  / L M ) )
6968oveq1d 5889 . . . . . . 7  |-  ( x  =  M  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L M
)  x.  ( A  / L 0 ) ) )
7069eqeq2d 2307 . . . . . 6  |-  ( x  =  M  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) ) )
7170rspcv 2893 . . . . 5  |-  ( M  e.  ZZ  ->  ( A. x  e.  ZZ  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L 0 ) )  ->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) ) )
7267, 41, 71sylc 56 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) )
7372adantr 451 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) )
74 oveq2 5882 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 10134 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9027 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2350 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 5890 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( A  / L
0 ) )
79 simpr 447 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  N  = 
0 )
8079oveq2d 5890 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L N )  =  ( A  / L
0 ) )
8180oveq2d 5890 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( ( A  / L M )  x.  ( A  / L N ) )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) )
8273, 78, 813eqtr4d 2338 . 2  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
83 lgsdi 20587 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
8419, 83syl3anl1 1230 . 2  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / L N ) ) )
8566, 82, 84pm2.61da2ne 2538 1  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L ( M  x.  N ) )  =  ( ( A  / L M )  x.  ( A  / 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   ifcif 3578   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    <_ cle 8884   2c2 9811   NN0cn0 9981   ZZcz 10040   ^cexp 11120    / Lclgs 20549
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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