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Theorem lgsdinn0 22691
Description: Variation on lgsdi 22683 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 990 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2 sq1 11972 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2453 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 10600 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 10617 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 9397 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 9875 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 11950 . . . . . . . . . . . . . . . . . 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 6118 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  ( 1  /L
x ) )
15 1lgs 22688 . . . . . . . . . . . . 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 2475 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L x )  =  1 )
1817oveq1d 6118 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  /L x )  x.  ( A  /L 0 ) )  =  ( 1  x.  ( A  /L 0 ) ) )
19 nn0z 10681 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10669 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 22661 . . . . . . . . . . . . 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 10760 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  e.  CC )
2524mulid2d 9416 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  /L 0 ) )  =  ( A  /L 0 ) )
2618, 25eqtr2d 2476 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
27 lgscl 22661 . . . . . . . . . . . . . 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 10760 . . . . . . . . . . . 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 9580 . . . . . . . . . 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 22660 . . . . . . . . . . . . 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 3813 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2495 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  0 )
3736oveq2d 6119 . . . . . . . . . 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 2486 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
3926, 38pm2.61dane 2701 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) ) )
4039ralrimiva 2811 . . . . . . 7  |-  ( A  e.  NN0  ->  A. x  e.  ZZ  ( A  /L 0 )  =  ( ( A  /L x )  x.  ( A  /L 0 ) ) )
41403ad2ant1 1009 . . . . . 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 6111 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  /L x )  =  ( A  /L N ) )
4342oveq1d 6118 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) )
4443eqeq2d 2454 . . . . . . 7  |-  ( x  =  N  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L N )  x.  ( A  /L 0 ) ) ) )
4544rspcv 3081 . . . . . 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 1009 . . . . . . . 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 10760 . . . . . 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 22661 . . . . . . . 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 10760 . . . . . 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 9419 . . . 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 2478 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L 0 )  =  ( ( A  /L 0 )  x.  ( A  /L
N ) ) )
58 oveq1 6110 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 10760 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9579 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2497 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 6119 . . 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 6119 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  /L M )  =  ( A  /L 0 ) )
6564oveq1d 6118 . . 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 2485 . 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 989 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
68 oveq2 6111 . . . . . . . 8  |-  ( x  =  M  ->  ( A  /L x )  =  ( A  /L M ) )
6968oveq1d 6118 . . . . . . 7  |-  ( x  =  M  ->  (
( A  /L
x )  x.  ( A  /L 0 ) )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) )
7069eqeq2d 2454 . . . . . 6  |-  ( x  =  M  ->  (
( A  /L 0 )  =  ( ( A  /L
x )  x.  ( A  /L 0 ) )  <->  ( A  /L 0 )  =  ( ( A  /L M )  x.  ( A  /L 0 ) ) ) )
7170rspcv 3081 . . . . 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 6111 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 10760 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9580 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2497 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 6119 . . 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 6119 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  /L N )  =  ( A  /L 0 ) )
8180oveq2d 6119 . . 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 2485 . 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 22683 . . 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 1266 . 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 2702 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   ifcif 3803   class class class wbr 4304  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    x. cmul 9299    <_ cle 9431   2c2 10383   NN0cn0 10591   ZZcz 10658   ^cexp 11877    /Lclgs 22645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-gcd 13703  df-prm 13776  df-phi 13853  df-pc 13916  df-lgs 22646
This theorem is referenced by: (None)
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