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Theorem lgsdinn0 21124
Description: Variation on lgsdi 21116 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 959 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2 sq1 11476 . . . . . . . . . . . . . . . 16  |-  ( 1 ^ 2 )  =  1
32eqeq2i 2446 . . . . . . . . . . . . . . 15  |-  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  ( A ^ 2 )  =  1 )
4 nn0re 10230 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  RR )
5 nn0ge0 10247 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  0  <_  A )
6 1re 9090 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
7 0le1 9551 . . . . . . . . . . . . . . . . . 18  |-  0  <_  1
8 sq11 11454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  0  <_  1 ) )  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  =  1
) )
96, 7, 8mpanr12 667 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
104, 5, 9syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  ( ( A ^ 2 )  =  ( 1 ^ 2 )  <->  A  = 
1 ) )
1110adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  ( 1 ^ 2 )  <-> 
A  =  1 ) )
123, 11syl5bbr 251 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( ( A ^
2 )  =  1  <-> 
A  =  1 ) )
1312biimpa 471 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  = 
1 )
1413oveq1d 6096 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  ( 1  / L
x ) )
15 1lgs 21121 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
1  / L x )  =  1 )
1615ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  / L x )  =  1 )
1714, 16eqtrd 2468 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L x )  =  1 )
1817oveq1d 6096 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( ( A  / L x )  x.  ( A  / L 0 ) )  =  ( 1  x.  ( A  / L
0 ) ) )
19 nn0z 10304 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2019ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  A  e.  ZZ )
21 0z 10293 . . . . . . . . . . . . 13  |-  0  e.  ZZ
22 lgscl 21094 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A  / L
0 )  e.  ZZ )
2320, 21, 22sylancl 644 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  e.  ZZ )
2423zcnd 10376 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  e.  CC )
2524mulid2d 9106 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( 1  x.  ( A  / L 0 ) )  =  ( A  / L 0 ) )
2618, 25eqtr2d 2469 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =  1 )  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
27 lgscl 21094 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  ZZ )
2819, 27sylan 458 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  ZZ )
2928zcnd 10376 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
x )  e.  CC )
3029adantr 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L x )  e.  CC )
3130mul01d 9265 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( ( A  / L x )  x.  0 )  =  0 )
3219adantr 452 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  A  e.  ZZ )
33 lgs0 21093 . . . . . . . . . . . . 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 3747 . . . . . . . . . . . 12  |-  ( ( A ^ 2 )  =/=  1  ->  if ( ( A ^
2 )  =  1 ,  1 ,  0 )  =  0 )
3634, 35sylan9eq 2488 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  0 )
3736oveq2d 6097 . . . . . . . . . 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 2479 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  x  e.  ZZ )  /\  ( A ^
2 )  =/=  1
)  ->  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
3926, 38pm2.61dane 2682 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  x  e.  ZZ )  ->  ( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) ) )
4039ralrimiva 2789 . . . . . . 7  |-  ( A  e.  NN0  ->  A. x  e.  ZZ  ( A  / L 0 )  =  ( ( A  / L x )  x.  ( A  / L
0 ) ) )
41403ad2ant1 978 . . . . . 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 6089 . . . . . . . . 9  |-  ( x  =  N  ->  ( A  / L x )  =  ( A  / L N ) )
4342oveq1d 6096 . . . . . . . 8  |-  ( x  =  N  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L N
)  x.  ( A  / L 0 ) ) )
4443eqeq2d 2447 . . . . . . 7  |-  ( x  =  N  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) ) )
4544rspcv 3048 . . . . . 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 58 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) )
4746adantr 452 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L N )  x.  ( A  / L 0 ) ) )
48193ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  A  e.  ZZ )
4948, 21, 22sylancl 644 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  e.  ZZ )
5049zcnd 10376 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  e.  CC )
5150adantr 452 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  e.  CC )
52 lgscl 21094 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
5348, 1, 52syl2anc 643 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
5453zcnd 10376 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  CC )
5554adantr 452 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L N )  e.  CC )
5651, 55mulcomd 9109 . . . 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 2471 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L 0 )  x.  ( A  / L N ) ) )
58 oveq1 6088 . . . . 5  |-  ( M  =  0  ->  ( M  x.  N )  =  ( 0  x.  N ) )
591zcnd 10376 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
6059mul02d 9264 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
0  x.  N )  =  0 )
6158, 60sylan9eqr 2490 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  x.  N )  =  0 )
6261oveq2d 6097 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( A  / L
0 ) )
63 simpr 448 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  M  = 
0 )
6463oveq2d 6097 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( A  / L M )  =  ( A  / L
0 ) )
6564oveq1d 6096 . . 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 2478 . 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 958 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
68 oveq2 6089 . . . . . . . 8  |-  ( x  =  M  ->  ( A  / L x )  =  ( A  / L M ) )
6968oveq1d 6096 . . . . . . 7  |-  ( x  =  M  ->  (
( A  / L
x )  x.  ( A  / L 0 ) )  =  ( ( A  / L M
)  x.  ( A  / L 0 ) ) )
7069eqeq2d 2447 . . . . . 6  |-  ( x  =  M  ->  (
( A  / L
0 )  =  ( ( A  / L
x )  x.  ( A  / L 0 ) )  <->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) ) )
7170rspcv 3048 . . . . 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 58 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) )
7372adantr 452 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L 0 )  =  ( ( A  / L M )  x.  ( A  / L 0 ) ) )
74 oveq2 6089 . . . . 5  |-  ( N  =  0  ->  ( M  x.  N )  =  ( M  x.  0 ) )
7567zcnd 10376 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
7675mul01d 9265 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  0 )  =  0 )
7774, 76sylan9eqr 2490 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( M  x.  N )  =  0 )
7877oveq2d 6097 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L ( M  x.  N ) )  =  ( A  / L
0 ) )
79 simpr 448 . . . . 5  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  N  = 
0 )
8079oveq2d 6097 . . . 4  |-  ( ( ( A  e.  NN0  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( A  / L N )  =  ( A  / L
0 ) )
8180oveq2d 6097 . . 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 2478 . 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 21116 . . 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 1232 . 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 2683 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   ifcif 3739   class class class wbr 4212  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    <_ cle 9121   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382    / Lclgs 21078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-phi 13155  df-pc 13211  df-lgs 21079
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