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Theorem modqmuladdnn0 10034
Description: Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
Assertion
Ref Expression
modqmuladdnn0  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) ) )
Distinct variable groups:    A, k    B, k    k, M

Proof of Theorem modqmuladdnn0
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  i  e.  ZZ )
21adantr 272 . . . . 5  |-  ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  < 
M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  -> 
i  e.  ZZ )
3 nn0cn 8891 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  A  e.  CC )
433ad2ant1 985 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  CC )
54ad2antrr 477 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  A  e.  CC )
6 nn0z 8978 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  ZZ )
7 zq 9320 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  A  e.  QQ )
86, 7syl 14 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  A  e.  QQ )
983ad2ant1 985 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  QQ )
109adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  QQ )
11 simpl2 968 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  QQ )
12 simpl3 969 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <  M )
1310, 11, 12modqcld 9994 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  QQ )
14 qcn 9328 . . . . . . . . . . . . 13  |-  ( ( A  mod  M )  e.  QQ  ->  ( A  mod  M )  e.  CC )
1513, 14syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  CC )
16 eleq1 2177 . . . . . . . . . . . . 13  |-  ( ( A  mod  M )  =  B  ->  (
( A  mod  M
)  e.  CC  <->  B  e.  CC ) )
1716adantl 273 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  mod  M )  e.  CC  <->  B  e.  CC ) )
1815, 17mpbid 146 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  CC )
1918adantr 272 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  B  e.  CC )
20 zcn 8963 . . . . . . . . . . . 12  |-  ( i  e.  ZZ  ->  i  e.  CC )
2120adantl 273 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  i  e.  CC )
22 qcn 9328 . . . . . . . . . . . . 13  |-  ( M  e.  QQ  ->  M  e.  CC )
2311, 22syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  CC )
2423adantr 272 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  CC )
2521, 24mulcld 7710 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
i  x.  M )  e.  CC )
265, 19, 25subadd2d 8015 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  B
)  =  ( i  x.  M )  <->  ( (
i  x.  M )  +  B )  =  A ) )
27 eqcom 2117 . . . . . . . . 9  |-  ( A  =  ( ( i  x.  M )  +  B )  <->  ( (
i  x.  M )  +  B )  =  A )
2826, 27syl6rbbr 198 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  =  ( (
i  x.  M )  +  B )  <->  ( A  -  B )  =  ( i  x.  M ) ) )
294adantr 272 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  CC )
3029, 18subcld 7996 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  -  B
)  e.  CC )
3130adantr 272 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  -  B )  e.  CC )
32 qre 9319 . . . . . . . . . . . 12  |-  ( M  e.  QQ  ->  M  e.  RR )
33323ad2ant2 986 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  RR )
3433ad2antrr 477 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  RR )
3512adantr 272 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  0  <  M )
3634, 35gt0ap0d 8309 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M #  0 )
3731, 21, 24, 36divmulap3d 8498 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( ( A  -  B )  /  M
)  =  i  <->  ( A  -  B )  =  ( i  x.  M ) ) )
38 oveq2 5736 . . . . . . . . . . . . . 14  |-  ( B  =  ( A  mod  M )  ->  ( A  -  B )  =  ( A  -  ( A  mod  M ) ) )
3938oveq1d 5743 . . . . . . . . . . . . 13  |-  ( B  =  ( A  mod  M )  ->  ( ( A  -  B )  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4039eqcoms 2118 . . . . . . . . . . . 12  |-  ( ( A  mod  M )  =  B  ->  (
( A  -  B
)  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M
) )
4140adantl 273 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  -  B )  /  M
)  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4241adantr 272 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  B
)  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M
) )
43 modqdiffl 10001 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
448, 43syl3an1 1232 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
4544ad2antrr 477 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
4642, 45eqtrd 2147 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  B
)  /  M )  =  ( |_ `  ( A  /  M
) ) )
4746eqeq1d 2123 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( ( A  -  B )  /  M
)  =  i  <->  ( |_ `  ( A  /  M
) )  =  i ) )
4828, 37, 473bitr2d 215 . . . . . . 7  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  =  ( (
i  x.  M )  +  B )  <->  ( |_ `  ( A  /  M
) )  =  i ) )
49 qre 9319 . . . . . . . . . . . 12  |-  ( A  e.  QQ  ->  A  e.  RR )
509, 49syl 14 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  RR )
51 nn0ge0 8906 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  0  <_  A )
52513ad2ant1 985 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  A )
53 simp3 966 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <  M )
54 divge0 8541 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( M  e.  RR  /\  0  <  M ) )  ->  0  <_  ( A  /  M ) )
5550, 52, 33, 53, 54syl22anc 1200 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  /  M
) )
56 simp2 965 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  QQ )
5753gt0ne0d 8193 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  =/=  0 )
58 qdivcl 9337 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
599, 56, 57, 58syl3anc 1199 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  /  M )  e.  QQ )
60 0z 8969 . . . . . . . . . . 11  |-  0  e.  ZZ
61 flqge 9948 . . . . . . . . . . 11  |-  ( ( ( A  /  M
)  e.  QQ  /\  0  e.  ZZ )  ->  ( 0  <_  ( A  /  M )  <->  0  <_  ( |_ `  ( A  /  M ) ) ) )
6259, 60, 61sylancl 407 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
0  <_  ( A  /  M )  <->  0  <_  ( |_ `  ( A  /  M ) ) ) )
6355, 62mpbid 146 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( |_ `  ( A  /  M ) ) )
64 breq2 3899 . . . . . . . . 9  |-  ( ( |_ `  ( A  /  M ) )  =  i  ->  (
0  <_  ( |_ `  ( A  /  M
) )  <->  0  <_  i ) )
6563, 64syl5ibcom 154 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( |_ `  ( A  /  M ) )  =  i  ->  0  <_  i ) )
6665ad2antrr 477 . . . . . . 7  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( |_ `  ( A  /  M ) )  =  i  ->  0  <_  i ) )
6748, 66sylbid 149 . . . . . 6  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  =  ( (
i  x.  M )  +  B )  -> 
0  <_  i )
)
6867imp 123 . . . . 5  |-  ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  < 
M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  -> 
0  <_  i )
69 elnn0z 8971 . . . . 5  |-  ( i  e.  NN0  <->  ( i  e.  ZZ  /\  0  <_ 
i ) )
702, 68, 69sylanbrc 411 . . . 4  |-  ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  < 
M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  -> 
i  e.  NN0 )
71 oveq1 5735 . . . . . . 7  |-  ( k  =  i  ->  (
k  x.  M )  =  ( i  x.  M ) )
7271oveq1d 5743 . . . . . 6  |-  ( k  =  i  ->  (
( k  x.  M
)  +  B )  =  ( ( i  x.  M )  +  B ) )
7372eqeq2d 2126 . . . . 5  |-  ( k  =  i  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( i  x.  M )  +  B
) ) )
7473adantl 273 . . . 4  |-  ( ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  /\  k  =  i )  ->  ( A  =  ( ( k  x.  M
)  +  B )  <-> 
A  =  ( ( i  x.  M )  +  B ) ) )
75 simpr 109 . . . 4  |-  ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  < 
M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  ->  A  =  ( (
i  x.  M )  +  B ) )
7670, 74, 75rspcedvd 2766 . . 3  |-  ( ( ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  < 
M )  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  /\  A  =  ( ( i  x.  M )  +  B ) )  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) )
77 modqmuladdim 10033 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) ) )
786, 77syl3an1 1232 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) ) )
7978imp 123 . . 3  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) )
8076, 79r19.29a 2549 . 2  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) )
8180ex 114 1  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463    =/= wne 2282   E.wrex 2391   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547    + caddc 7550    x. cmul 7552    < clt 7724    <_ cle 7725    - cmin 7856    / cdiv 8345   NN0cn0 8881   ZZcz 8958   QQcq 9313   |_cfl 9934    mod cmo 9988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-ico 9570  df-fl 9936  df-mod 9989
This theorem is referenced by: (None)
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