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Theorem modqmuladdnn0 10096
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 274 . . . . 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 8945 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  A  e.  CC )
433ad2ant1 987 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  CC )
54ad2antrr 479 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  A  e.  CC )
6 nn0z 9032 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  ZZ )
7 zq 9374 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  A  e.  QQ )
86, 7syl 14 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  A  e.  QQ )
983ad2ant1 987 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  QQ )
109adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  QQ )
11 simpl2 970 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  QQ )
12 simpl3 971 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <  M )
1310, 11, 12modqcld 10056 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  QQ )
14 qcn 9382 . . . . . . . . . . . . 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 2180 . . . . . . . . . . . . 13  |-  ( ( A  mod  M )  =  B  ->  (
( A  mod  M
)  e.  CC  <->  B  e.  CC ) )
1716adantl 275 . . . . . . . . . . . 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 274 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  B  e.  CC )
20 zcn 9017 . . . . . . . . . . . 12  |-  ( i  e.  ZZ  ->  i  e.  CC )
2120adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  i  e.  CC )
22 qcn 9382 . . . . . . . . . . . . 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 274 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  CC )
2521, 24mulcld 7754 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
i  x.  M )  e.  CC )
265, 19, 25subadd2d 8060 . . . . . . . . 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 2119 . . . . . . . . 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 274 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  CC )
3029, 18subcld 8041 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  -  B
)  e.  CC )
3130adantr 274 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  -  B )  e.  CC )
32 qre 9373 . . . . . . . . . . . 12  |-  ( M  e.  QQ  ->  M  e.  RR )
33323ad2ant2 988 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  RR )
3433ad2antrr 479 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  RR )
3512adantr 274 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  0  <  M )
3634, 35gt0ap0d 8358 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M #  0 )
3731, 21, 24, 36divmulap3d 8552 . . . . . . . 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 5750 . . . . . . . . . . . . . 14  |-  ( B  =  ( A  mod  M )  ->  ( A  -  B )  =  ( A  -  ( A  mod  M ) ) )
3938oveq1d 5757 . . . . . . . . . . . . 13  |-  ( B  =  ( A  mod  M )  ->  ( ( A  -  B )  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4039eqcoms 2120 . . . . . . . . . . . 12  |-  ( ( A  mod  M )  =  B  ->  (
( A  -  B
)  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M
) )
4140adantl 275 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  -  B )  /  M
)  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4241adantr 274 . . . . . . . . . 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 10063 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
448, 43syl3an1 1234 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
4544ad2antrr 479 . . . . . . . . . 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 2150 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  B
)  /  M )  =  ( |_ `  ( A  /  M
) ) )
4746eqeq1d 2126 . . . . . . . 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 9373 . . . . . . . . . . . 12  |-  ( A  e.  QQ  ->  A  e.  RR )
509, 49syl 14 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  RR )
51 nn0ge0 8960 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  0  <_  A )
52513ad2ant1 987 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  A )
53 simp3 968 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <  M )
54 divge0 8595 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( M  e.  RR  /\  0  <  M ) )  ->  0  <_  ( A  /  M ) )
5550, 52, 33, 53, 54syl22anc 1202 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  /  M
) )
56 simp2 967 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  QQ )
5753gt0ne0d 8242 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  =/=  0 )
58 qdivcl 9391 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
599, 56, 57, 58syl3anc 1201 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  /  M )  e.  QQ )
60 0z 9023 . . . . . . . . . . 11  |-  0  e.  ZZ
61 flqge 10010 . . . . . . . . . . 11  |-  ( ( ( A  /  M
)  e.  QQ  /\  0  e.  ZZ )  ->  ( 0  <_  ( A  /  M )  <->  0  <_  ( |_ `  ( A  /  M ) ) ) )
6259, 60, 61sylancl 409 . . . . . . . . . 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 3903 . . . . . . . . 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 479 . . . . . . 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 9025 . . . . 5  |-  ( i  e.  NN0  <->  ( i  e.  ZZ  /\  0  <_ 
i ) )
702, 68, 69sylanbrc 413 . . . 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 5749 . . . . . . 7  |-  ( k  =  i  ->  (
k  x.  M )  =  ( i  x.  M ) )
7271oveq1d 5757 . . . . . 6  |-  ( k  =  i  ->  (
( k  x.  M
)  +  B )  =  ( ( i  x.  M )  +  B ) )
7372eqeq2d 2129 . . . . 5  |-  ( k  =  i  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( i  x.  M )  +  B
) ) )
7473adantl 275 . . . 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 2769 . . 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 10095 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) ) )
786, 77syl3an1 1234 . . . 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 2552 . 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 947    = wceq 1316    e. wcel 1465    =/= wne 2285   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901    / cdiv 8399   NN0cn0 8935   ZZcz 9012   QQcq 9367   |_cfl 9996    mod cmo 10050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-n0 8936  df-z 9013  df-q 9368  df-rp 9398  df-ico 9632  df-fl 9998  df-mod 10051
This theorem is referenced by: (None)
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