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Theorem lgsdinn0 24131
Description: Variation on lgsdi 24123 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 1007 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2 sq1 12366 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2447 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 10878 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 10895 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 9641 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 10136 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 12344 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  0  <_  1 ) )  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  =  1
) )
96, 7, 8mpanr12 689 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
104, 5, 9syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  = 
1 ) )
1110adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
123, 11syl5bbr 262 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  1  <-> 
A  =  1 ) )
1312biimpa 486 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  = 
1 )
1413oveq1d 6320 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  ( 1  /L
x ) )
15 1lgs 24128 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
1  /L x )  =  1 )
1615ad2antlr 731 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  /L x )  =  1 )
1714, 16eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  1 )
1817oveq1d 6320 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  /L x )  x.  ( A  /L 0 ) )  =  ( 1  x.  ( A  /L 0 ) ) )
19 nn0z 10960 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10948 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 24101 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A  /L 0 )  e.  ZZ )
2320, 21, 22sylancl 666 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  e.  ZZ )
2423zcnd 11041 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  e.  CC )
2524mulid2d 9660 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  /L 0 ) )  =  ( A  /L 0 ) )
2618, 25eqtr2d 2471 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
27 lgscl 24101 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  ZZ )
2819, 27sylan 473 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  ZZ )
2928zcnd 11041 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L
x )  e.  CC )
3029adantr 466 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L x )  e.  CC )
3130mul01d 9831 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( ( A  /L x )  x.  0 )  =  0 )
3219adantr 466 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  A  e.  ZZ )
33 lgs0 24100 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A  /L 0 )  =  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) )
3432, 33syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L 0 )  =  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) )
35 ifnefalse 3927 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2490 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  0 )
3736oveq2d 6321 . . . . . . . . . 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 2481 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
3926, 38pm2.61dane 2749 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) ) )
4039ralrimiva 2846 . . . . . . 7  |-  ( A  e.  NN0  ->  A. x  e.  ZZ  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
41403ad2ant1 1026 . . . . . 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 6313 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  /L x )  =  ( A  /L N ) )
4342oveq1d 6320 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
4443eqeq2d 2443 . . . . . . 7  |-  ( x  =  N  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) ) )
4544rspcv 3184 . . . . . 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 62 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
4746adantr 466 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
48193ad2ant1 1026 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  A  e.  ZZ )
4948, 21, 22sylancl 666 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  e.  ZZ )
5049zcnd 11041 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  e.  CC )
5150adantr 466 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  e.  CC )
52 lgscl 24101 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L
N )  e.  ZZ )
5348, 1, 52syl2anc 665 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L N )  e.  ZZ )
5453zcnd 11041 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L N )  e.  CC )
5554adantr 466 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L N )  e.  CC )
5651, 55mulcomd 9663 . . . 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 2473 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L 0 )  x.  ( A  /L
N ) ) )
58 oveq1 6312 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 11041 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9830 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2492 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 6321 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L ( M  x.  N ) )  =  ( A  /L 0 ) )
63 simpr 462 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
6463oveq2d 6321 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L M )  =  ( A  /L 0 ) )
6564oveq1d 6320 . . 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 2480 . 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 1006 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
68 oveq2 6313 . . . . . . . 8  |-  ( x  =  M  ->  ( A  /L x )  =  ( A  /L M ) )
6968oveq1d 6320 . . . . . . 7  |-  ( x  =  M  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
7069eqeq2d 2443 . . . . . 6  |-  ( x  =  M  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) ) )
7170rspcv 3184 . . . . 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 62 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
7372adantr 466 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
74 oveq2 6313 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 11041 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9831 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2492 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 6321 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L ( M  x.  N ) )  =  ( A  /L 0 ) )
79 simpr 462 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  N  = 
0 )
8079oveq2d 6321 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L N )  =  ( A  /L 0 ) )
8180oveq2d 6321 . . 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 2480 . 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 24123 . . 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 1312 . 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 2750 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   ifcif 3915   class class class wbr 4426  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    <_ cle 9675   2c2 10659   NN0cn0 10869   ZZcz 10937   ^cexp 12269    /Lclgs 24085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594  df-phi 14683  df-pc 14750  df-lgs 24086
This theorem is referenced by: (None)
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