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Theorem lgsdinn0 23741
Description: Variation on lgsdi 23733 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 998 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2 sq1 12265 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2475 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 10825 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 10842 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 9612 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 10097 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 12243 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  0  <_  1 ) )  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  =  1
) )
96, 7, 8mpanr12 685 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
104, 5, 9syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  = 
1 ) )
1110adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
123, 11syl5bbr 259 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  1  <-> 
A  =  1 ) )
1312biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  = 
1 )
1413oveq1d 6311 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  ( 1  /L
x ) )
15 1lgs 23738 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
1  /L x )  =  1 )
1615ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  /L x )  =  1 )
1714, 16eqtrd 2498 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  1 )
1817oveq1d 6311 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  /L x )  x.  ( A  /L 0 ) )  =  ( 1  x.  ( A  /L 0 ) ) )
19 nn0z 10908 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10896 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 23711 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A  /L 0 )  e.  ZZ )
2320, 21, 22sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  e.  ZZ )
2423zcnd 10991 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  e.  CC )
2524mulid2d 9631 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  /L 0 ) )  =  ( A  /L 0 ) )
2618, 25eqtr2d 2499 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
27 lgscl 23711 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  ZZ )
2819, 27sylan 471 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  ZZ )
2928zcnd 10991 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  CC )
3029adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L x )  e.  CC )
3130mul01d 9796 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( ( A  /L x )  x.  0 )  =  0 )
3219adantr 465 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  A  e.  ZZ )
33 lgs0 23710 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
3432, 33syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L 0 )  =  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) )
35 ifnefalse 3956 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2518 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  0 )
3736oveq2d 6312 . . . . . . . . . 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 2509 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
3926, 38pm2.61dane 2775 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) ) )
4039ralrimiva 2871 . . . . . . 7  |-  ( A  e.  NN0  ->  A. x  e.  ZZ  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
41403ad2ant1 1017 . . . . . 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 6304 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  /L x )  =  ( A  /L N ) )
4342oveq1d 6311 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
4443eqeq2d 2471 . . . . . . 7  |-  ( x  =  N  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) ) )
4544rspcv 3206 . . . . . 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 60 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
4746adantr 465 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
48193ad2ant1 1017 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  A  e.  ZZ )
4948, 21, 22sylancl 662 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  e.  ZZ )
5049zcnd 10991 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  e.  CC )
5150adantr 465 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  e.  CC )
52 lgscl 23711 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L
N )  e.  ZZ )
5348, 1, 52syl2anc 661 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L N )  e.  ZZ )
5453zcnd 10991 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L N )  e.  CC )
5554adantr 465 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L N )  e.  CC )
5651, 55mulcomd 9634 . . . 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 2501 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L 0 )  x.  ( A  /L
N ) ) )
58 oveq1 6303 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 10991 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9795 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2520 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 6312 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L ( M  x.  N ) )  =  ( A  /L 0 ) )
63 simpr 461 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
6463oveq2d 6312 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L M )  =  ( A  /L 0 ) )
6564oveq1d 6311 . . 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 2508 . 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 997 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
68 oveq2 6304 . . . . . . . 8  |-  ( x  =  M  ->  ( A  /L x )  =  ( A  /L M ) )
6968oveq1d 6311 . . . . . . 7  |-  ( x  =  M  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
7069eqeq2d 2471 . . . . . 6  |-  ( x  =  M  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) ) )
7170rspcv 3206 . . . . 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 60 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
7372adantr 465 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
74 oveq2 6304 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 10991 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9796 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2520 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 6312 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L ( M  x.  N ) )  =  ( A  /L 0 ) )
79 simpr 461 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  N  = 
0 )
8079oveq2d 6312 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L N )  =  ( A  /L 0 ) )
8180oveq2d 6312 . . 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 2508 . 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 23733 . . 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 1276 . 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 2776 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ifcif 3944   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    <_ cle 9646   2c2 10606   NN0cn0 10816   ZZcz 10885   ^cexp 12169    /Lclgs 23695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-phi 14308  df-pc 14373  df-lgs 23696
This theorem is referenced by: (None)
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