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Theorem dvdsflf1o 20427
Description: A bijection from the numbers less than  N  /  A to the multiples of  A less than  N. Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
dvdsflf1o.1  |-  ( ph  ->  A  e.  RR )
dvdsflf1o.2  |-  ( ph  ->  N  e.  NN )
dvdsflf1o.f  |-  F  =  ( n  e.  ( 1 ... ( |_
`  ( A  /  N ) ) ) 
|->  ( N  x.  n
) )
Assertion
Ref Expression
dvdsflf1o  |-  ( ph  ->  F : ( 1 ... ( |_ `  ( A  /  N
) ) ) -1-1-onto-> { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x } )
Distinct variable groups:    x, n, A    n, N, x    ph, n
Allowed substitution hints:    ph( x)    F( x, n)

Proof of Theorem dvdsflf1o
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 dvdsflf1o.f . 2  |-  F  =  ( n  e.  ( 1 ... ( |_
`  ( A  /  N ) ) ) 
|->  ( N  x.  n
) )
2 dvdsflf1o.2 . . . . 5  |-  ( ph  ->  N  e.  NN )
3 elfznn 10819 . . . . 5  |-  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  ->  n  e.  NN )
4 nnmulcl 9769 . . . . 5  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( N  x.  n
)  e.  NN )
52, 3, 4syl2an 463 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  e.  NN )
6 dvdsflf1o.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
76, 2nndivred 9794 . . . . . . . 8  |-  ( ph  ->  ( A  /  N
)  e.  RR )
8 fznnfl 10966 . . . . . . . 8  |-  ( ( A  /  N )  e.  RR  ->  (
n  e.  ( 1 ... ( |_ `  ( A  /  N
) ) )  <->  ( n  e.  NN  /\  n  <_ 
( A  /  N
) ) ) )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  ( n  e.  ( 1 ... ( |_
`  ( A  /  N ) ) )  <-> 
( n  e.  NN  /\  n  <_  ( A  /  N ) ) ) )
109simplbda 607 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  n  <_  ( A  /  N ) )
113adantl 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  n  e.  NN )
1211nnred 9761 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  n  e.  RR )
136adantr 451 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  A  e.  RR )
142nnred 9761 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
1514adantr 451 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  N  e.  RR )
162nngt0d 9789 . . . . . . . 8  |-  ( ph  ->  0  <  N )
1716adantr 451 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  0  <  N )
18 lemuldiv2 9636 . . . . . . 7  |-  ( ( n  e.  RR  /\  A  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( N  x.  n )  <_  A  <->  n  <_  ( A  /  N ) ) )
1912, 13, 15, 17, 18syl112anc 1186 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( ( N  x.  n )  <_  A  <->  n  <_  ( A  /  N ) ) )
2010, 19mpbird 223 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  <_  A
)
212nnzd 10116 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
22 elfzelz 10798 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  ->  n  e.  ZZ )
23 zmulcl 10066 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  ->  ( N  x.  n
)  e.  ZZ )
2421, 22, 23syl2an 463 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  e.  ZZ )
25 flge 10937 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  x.  n
)  e.  ZZ )  ->  ( ( N  x.  n )  <_  A 
<->  ( N  x.  n
)  <_  ( |_ `  A ) ) )
2613, 24, 25syl2anc 642 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( ( N  x.  n )  <_  A  <->  ( N  x.  n )  <_  ( |_ `  A ) ) )
2720, 26mpbid 201 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  <_  ( |_ `  A ) )
286flcld 10930 . . . . . 6  |-  ( ph  ->  ( |_ `  A
)  e.  ZZ )
2928adantr 451 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( |_ `  A )  e.  ZZ )
30 fznn 10852 . . . . 5  |-  ( ( |_ `  A )  e.  ZZ  ->  (
( N  x.  n
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( ( N  x.  n )  e.  NN  /\  ( N  x.  n )  <_ 
( |_ `  A
) ) ) )
3129, 30syl 15 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( ( N  x.  n )  e.  ( 1 ... ( |_ `  A ) )  <-> 
( ( N  x.  n )  e.  NN  /\  ( N  x.  n
)  <_  ( |_ `  A ) ) ) )
325, 27, 31mpbir2and 888 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  e.  ( 1 ... ( |_
`  A ) ) )
33 dvdsmul1 12550 . . . 4  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  ->  N  ||  ( N  x.  n ) )
3421, 22, 33syl2an 463 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  N  ||  ( N  x.  n )
)
35 breq2 4027 . . . 4  |-  ( x  =  ( N  x.  n )  ->  ( N  ||  x  <->  N  ||  ( N  x.  n )
) )
3635elrab 2923 . . 3  |-  ( ( N  x.  n )  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x } 
<->  ( ( N  x.  n )  e.  ( 1 ... ( |_
`  A ) )  /\  N  ||  ( N  x.  n )
) )
3732, 34, 36sylanbrc 645 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  ( N  x.  n )  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)
38 breq2 4027 . . . . . . 7  |-  ( x  =  m  ->  ( N  ||  x  <->  N  ||  m
) )
3938elrab 2923 . . . . . 6  |-  ( m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x } 
<->  ( m  e.  ( 1 ... ( |_
`  A ) )  /\  N  ||  m
) )
4039simprbi 450 . . . . 5  |-  ( m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }  ->  N  ||  m
)
4140adantl 452 . . . 4  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  N  ||  m
)
4239simplbi 446 . . . . . . 7  |-  ( m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }  ->  m  e.  ( 1 ... ( |_
`  A ) ) )
4342adantl 452 . . . . . 6  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  m  e.  ( 1 ... ( |_ `  A ) ) )
44 elfznn 10819 . . . . . 6  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
4543, 44syl 15 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  m  e.  NN )
462adantr 451 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  N  e.  NN )
47 nndivdvds 12537 . . . . 5  |-  ( ( m  e.  NN  /\  N  e.  NN )  ->  ( N  ||  m  <->  ( m  /  N )  e.  NN ) )
4845, 46, 47syl2anc 642 . . . 4  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( N  ||  m  <->  ( m  /  N )  e.  NN ) )
4941, 48mpbid 201 . . 3  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( m  /  N )  e.  NN )
50 fznnfl 10966 . . . . . . 7  |-  ( A  e.  RR  ->  (
m  e.  ( 1 ... ( |_ `  A ) )  <->  ( m  e.  NN  /\  m  <_  A ) ) )
516, 50syl 15 . . . . . 6  |-  ( ph  ->  ( m  e.  ( 1 ... ( |_
`  A ) )  <-> 
( m  e.  NN  /\  m  <_  A )
) )
5251simplbda 607 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  <_  A )
5342, 52sylan2 460 . . . 4  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  m  <_  A )
5445nnred 9761 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  m  e.  RR )
556adantr 451 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  A  e.  RR )
5614adantr 451 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  N  e.  RR )
5716adantr 451 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  0  <  N )
58 lediv1 9621 . . . . 5  |-  ( ( m  e.  RR  /\  A  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( m  <_  A  <->  ( m  /  N )  <_  ( A  /  N ) ) )
5954, 55, 56, 57, 58syl112anc 1186 . . . 4  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( m  <_  A  <->  ( m  /  N )  <_  ( A  /  N ) ) )
6053, 59mpbid 201 . . 3  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( m  /  N )  <_  ( A  /  N ) )
617adantr 451 . . . 4  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( A  /  N )  e.  RR )
62 fznnfl 10966 . . . 4  |-  ( ( A  /  N )  e.  RR  ->  (
( m  /  N
)  e.  ( 1 ... ( |_ `  ( A  /  N
) ) )  <->  ( (
m  /  N )  e.  NN  /\  (
m  /  N )  <_  ( A  /  N ) ) ) )
6361, 62syl 15 . . 3  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( (
m  /  N )  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  <->  ( ( m  /  N )  e.  NN  /\  ( m  /  N )  <_ 
( A  /  N
) ) ) )
6449, 60, 63mpbir2and 888 . 2  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  ( m  /  N )  e.  ( 1 ... ( |_
`  ( A  /  N ) ) ) )
6545nncnd 9762 . . . . 5  |-  ( (
ph  /\  m  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
)  ->  m  e.  CC )
6665adantrl 696 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  m  e.  CC )
672nncnd 9762 . . . . 5  |-  ( ph  ->  N  e.  CC )
6867adantr 451 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  N  e.  CC )
6911nncnd 9762 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) ) )  ->  n  e.  CC )
7069adantrr 697 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  n  e.  CC )
712nnne0d 9790 . . . . 5  |-  ( ph  ->  N  =/=  0 )
7271adantr 451 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  N  =/=  0 )
7366, 68, 70, 72divmuld 9558 . . 3  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  (
( m  /  N
)  =  n  <->  ( N  x.  n )  =  m ) )
74 eqcom 2285 . . 3  |-  ( n  =  ( m  /  N )  <->  ( m  /  N )  =  n )
75 eqcom 2285 . . 3  |-  ( m  =  ( N  x.  n )  <->  ( N  x.  n )  =  m )
7673, 74, 753bitr4g 279 . 2  |-  ( (
ph  /\  ( n  e.  ( 1 ... ( |_ `  ( A  /  N ) ) )  /\  m  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x }
) )  ->  (
n  =  ( m  /  N )  <->  m  =  ( N  x.  n
) ) )
771, 37, 64, 76f1o2d 6069 1  |-  ( ph  ->  F : ( 1 ... ( |_ `  ( A  /  N
) ) ) -1-1-onto-> { x  e.  ( 1 ... ( |_ `  A ) )  |  N  ||  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   class class class wbr 4023    e. cmpt 4077   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   |_cfl 10924    || cdivides 12531
This theorem is referenced by:  dvdsflsumcom  20428  logfac2  20456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fl 10925  df-dvds 12532
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