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Theorem lgsdinn0 20579
Description: Variation on lgsdi 20571 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 11198 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2293 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 9974 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 9991 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 8837 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 9297 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 11176 . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  ( 1  / L
x ) )
15 1lgs 20576 . . . . . . . . . . . . 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 2315 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  1 )
1817oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  / L x )  x.  ( A  / L 0 ) )  =  ( 1  x.  ( A  / L
0 ) ) )
19 nn0z 10046 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10035 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 20549 . . . . . . . . . . . . 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 10118 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  e.  CC )
2524mulid2d 8853 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  / L 0 ) )  =  ( A  / L 0 ) )
2618, 25eqtr2d 2316 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
27 lgscl 20549 . . . . . . . . . . . . . 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 10118 . . . . . . . . . . . 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 9011 . . . . . . . . . 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 20548 . . . . . . . . . . . . 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 3573 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2335 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  0 )
3736oveq2d 5874 . . . . . . . . . 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 2326 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
3926, 38pm2.61dane 2524 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) ) )
4039ralrimiva 2626 . . . . . . 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 5866 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  / L x )  =  ( A  / L N ) )
4342oveq1d 5873 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L N
)  x.  ( A  / L 0 ) ) )
4443eqeq2d 2294 . . . . . . 7  |-  ( x  =  N  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) ) )
4544rspcv 2880 . . . . . 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 10118 . . . . . 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 20549 . . . . . . . 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 10118 . . . . . 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 8856 . . . 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 2318 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L 0 )  x.  ( A  / L N ) ) )
58 oveq1 5865 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 10118 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9010 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2337 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 5874 . . 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 5874 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L M )  =  ( A  / L
0 ) )
6564oveq1d 5873 . . 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 2325 . 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 5866 . . . . . . . 8  |-  ( x  =  M  ->  ( A  / L x )  =  ( A  / L M ) )
6968oveq1d 5873 . . . . . . 7  |-  ( x  =  M  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L M
)  x.  ( A  / L 0 ) ) )
7069eqeq2d 2294 . . . . . 6  |-  ( x  =  M  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) ) )
7170rspcv 2880 . . . . 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 5866 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 10118 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9011 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2337 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 5874 . . 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 5874 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L N )  =  ( A  / L
0 ) )
8180oveq2d 5874 . . 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 2325 . 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 20571 . . 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 2525 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104    / Lclgs 20533
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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