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Theorem modqmuladdnn0 10398
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 110 . . . . . 6  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  i  e.  ZZ )
21adantr 276 . . . . 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 eqcom 2191 . . . . . . . . 9  |-  ( A  =  ( ( i  x.  M )  +  B )  <->  ( (
i  x.  M )  +  B )  =  A )
4 nn0cn 9215 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  A  e.  CC )
543ad2ant1 1020 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  CC )
65ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  A  e.  CC )
7 nn0z 9302 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  NN0  ->  A  e.  ZZ )
8 zq 9655 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  ZZ  ->  A  e.  QQ )
97, 8syl 14 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN0  ->  A  e.  QQ )
1093ad2ant1 1020 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  QQ )
1110adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  QQ )
12 simpl2 1003 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  QQ )
13 simpl3 1004 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <  M )
1411, 12, 13modqcld 10358 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  QQ )
15 qcn 9663 . . . . . . . . . . . . 13  |-  ( ( A  mod  M )  e.  QQ  ->  ( A  mod  M )  e.  CC )
1614, 15syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  CC )
17 eleq1 2252 . . . . . . . . . . . . 13  |-  ( ( A  mod  M )  =  B  ->  (
( A  mod  M
)  e.  CC  <->  B  e.  CC ) )
1817adantl 277 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  mod  M )  e.  CC  <->  B  e.  CC ) )
1916, 18mpbid 147 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  CC )
2019adantr 276 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  B  e.  CC )
21 zcn 9287 . . . . . . . . . . . 12  |-  ( i  e.  ZZ  ->  i  e.  CC )
2221adantl 277 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  i  e.  CC )
23 qcn 9663 . . . . . . . . . . . . 13  |-  ( M  e.  QQ  ->  M  e.  CC )
2412, 23syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  CC )
2524adantr 276 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  CC )
2622, 25mulcld 8007 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
i  x.  M )  e.  CC )
276, 20, 26subadd2d 8316 . . . . . . . . 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 ) )
283, 27bitr4id 199 . . . . . . . 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 ) ) )
295adantr 276 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  CC )
3029, 19subcld 8297 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  -  B
)  e.  CC )
3130adantr 276 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  -  B )  e.  CC )
32 qre 9654 . . . . . . . . . . . 12  |-  ( M  e.  QQ  ->  M  e.  RR )
33323ad2ant2 1021 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  RR )
3433ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M  e.  RR )
3513adantr 276 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  0  <  M )
3634, 35gt0ap0d 8615 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  M #  0 )
3731, 22, 25, 36divmulap3d 8811 . . . . . . . 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 5903 . . . . . . . . . . . . . 14  |-  ( B  =  ( A  mod  M )  ->  ( A  -  B )  =  ( A  -  ( A  mod  M ) ) )
3938oveq1d 5910 . . . . . . . . . . . . 13  |-  ( B  =  ( A  mod  M )  ->  ( ( A  -  B )  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4039eqcoms 2192 . . . . . . . . . . . 12  |-  ( ( A  mod  M )  =  B  ->  (
( A  -  B
)  /  M )  =  ( ( A  -  ( A  mod  M ) )  /  M
) )
4140adantl 277 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  -  B )  /  M
)  =  ( ( A  -  ( A  mod  M ) )  /  M ) )
4241adantr 276 . . . . . . . . . 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 10365 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
449, 43syl3an1 1282 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  -  ( A  mod  M ) )  /  M )  =  ( |_ `  ( A  /  M ) ) )
4544ad2antrr 488 . . . . . . . . . 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 2222 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( A  -  B
)  /  M )  =  ( |_ `  ( A  /  M
) ) )
4746eqeq1d 2198 . . . . . . . 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 216 . . . . . . 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 9654 . . . . . . . . . . . 12  |-  ( A  e.  QQ  ->  A  e.  RR )
5010, 49syl 14 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  A  e.  RR )
51 nn0ge0 9230 . . . . . . . . . . . 12  |-  ( A  e.  NN0  ->  0  <_  A )
52513ad2ant1 1020 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  A )
53 simp3 1001 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <  M )
54 divge0 8859 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( M  e.  RR  /\  0  <  M ) )  ->  0  <_  ( A  /  M ) )
5550, 52, 33, 53, 54syl22anc 1250 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  /  M
) )
56 simp2 1000 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  QQ )
5753gt0ne0d 8498 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  M  =/=  0 )
58 qdivcl 9672 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
5910, 56, 57, 58syl3anc 1249 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  /  M )  e.  QQ )
60 0z 9293 . . . . . . . . . . 11  |-  0  e.  ZZ
61 flqge 10312 . . . . . . . . . . 11  |-  ( ( ( A  /  M
)  e.  QQ  /\  0  e.  ZZ )  ->  ( 0  <_  ( A  /  M )  <->  0  <_  ( |_ `  ( A  /  M ) ) ) )
6259, 60, 61sylancl 413 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
0  <_  ( A  /  M )  <->  0  <_  ( |_ `  ( A  /  M ) ) ) )
6355, 62mpbid 147 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( |_ `  ( A  /  M ) ) )
64 breq2 4022 . . . . . . . . 9  |-  ( ( |_ `  ( A  /  M ) )  =  i  ->  (
0  <_  ( |_ `  ( A  /  M
) )  <->  0  <_  i ) )
6563, 64syl5ibcom 155 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( |_ `  ( A  /  M ) )  =  i  ->  0  <_  i ) )
6665ad2antrr 488 . . . . . . 7  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  (
( |_ `  ( A  /  M ) )  =  i  ->  0  <_  i ) )
6748, 66sylbid 150 . . . . . 6  |-  ( ( ( ( A  e. 
NN0  /\  M  e.  QQ  /\  0  <  M
)  /\  ( A  mod  M )  =  B )  /\  i  e.  ZZ )  ->  ( A  =  ( (
i  x.  M )  +  B )  -> 
0  <_  i )
)
6867imp 124 . . . . 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 9295 . . . . 5  |-  ( i  e.  NN0  <->  ( i  e.  ZZ  /\  0  <_ 
i ) )
702, 68, 69sylanbrc 417 . . . 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 5902 . . . . . . 7  |-  ( k  =  i  ->  (
k  x.  M )  =  ( i  x.  M ) )
7271oveq1d 5910 . . . . . 6  |-  ( k  =  i  ->  (
( k  x.  M
)  +  B )  =  ( ( i  x.  M )  +  B ) )
7372eqeq2d 2201 . . . . 5  |-  ( k  =  i  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( i  x.  M )  +  B
) ) )
7473adantl 277 . . . 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 110 . . . 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 2862 . . 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 10397 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) ) )
787, 77syl3an1 1282 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. i  e.  ZZ  A  =  ( (
i  x.  M )  +  B ) ) )
7978imp 124 . . 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 2633 . 2  |-  ( ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) )
8180ex 115 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    =/= wne 2360   E.wrex 2469   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840    + caddc 7843    x. cmul 7845    < clt 8021    <_ cle 8022    - cmin 8157    / cdiv 8658   NN0cn0 9205   ZZcz 9282   QQcq 9648   |_cfl 10298    mod cmo 10352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-ico 9923  df-fl 10300  df-mod 10353
This theorem is referenced by: (None)
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