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Theorem mptfzshft 11469
Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
Hypotheses
Ref Expression
mptfzshft.1  |-  ( ph  ->  K  e.  ZZ )
mptfzshft.2  |-  ( ph  ->  M  e.  ZZ )
mptfzshft.3  |-  ( ph  ->  N  e.  ZZ )
Assertion
Ref Expression
mptfzshft  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
Distinct variable groups:    j, K    j, M    j, N    ph, j

Proof of Theorem mptfzshft
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 eqid 2189 . 2  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
2 elfzelz 10044 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
32adantl 277 . . 3  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  j  e.  ZZ )
4 mptfzshft.1 . . . 4  |-  ( ph  ->  K  e.  ZZ )
54adantr 276 . . 3  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  K  e.  ZZ )
63, 5zsubcld 9399 . 2  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
j  -  K )  e.  ZZ )
7 elfzelz 10044 . . . 4  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
87adantl 277 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  k  e.  ZZ )
94adantr 276 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  K  e.  ZZ )
108, 9zaddcld 9398 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( k  +  K )  e.  ZZ )
11 simprr 531 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
1211oveq1d 5906 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
132ad2antrl 490 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
144adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
15 zcn 9277 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
16 zcn 9277 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
17 npcan 8185 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  K  e.  CC )  ->  ( ( j  -  K )  +  K
)  =  j )
1815, 16, 17syl2an 289 . . . . . . . 8  |-  ( ( j  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( j  -  K )  +  K
)  =  j )
1913, 14, 18syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
2012, 19eqtr2d 2223 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
21 simprl 529 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2220, 21eqeltrrd 2267 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
23 mptfzshft.2 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2423adantr 276 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
25 mptfzshft.3 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2625adantr 276 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
2713, 14zsubcld 9399 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
2811, 27eqeltrd 2266 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
29 fzaddel 10078 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3024, 26, 28, 14, 29syl22anc 1250 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3122, 30mpbird 167 . . . 4  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ( M ... N ) )
3231, 20jca 306 . . 3  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
33 simprr 531 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
34 simprl 529 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3523adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3625adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
377ad2antrl 490 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
384adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
3935, 36, 37, 38, 29syl22anc 1250 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
4034, 39mpbid 147 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4133, 40eqeltrd 2266 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4233oveq1d 5906 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
43 zcn 9277 . . . . . . 7  |-  ( k  e.  ZZ  ->  k  e.  CC )
44 pncan 8182 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  +  K )  -  K
)  =  k )
4543, 16, 44syl2an 289 . . . . . 6  |-  ( ( k  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( k  +  K )  -  K
)  =  k )
4637, 38, 45syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
4742, 46eqtr2d 2223 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
4841, 47jca 306 . . 3  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
4932, 48impbida 596 . 2  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
501, 6, 10, 49f1od 6092 1  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    |-> cmpt 4079   -1-1-onto->wf1o 5230  (class class class)co 5891   CCcc 7828    + caddc 7833    - cmin 8147   ZZcz 9272   ...cfz 10027
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 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-ltadd 7946
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-rab 2477  df-v 2754  df-sbc 2978  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-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-n0 9196  df-z 9273  df-uz 9548  df-fz 10028
This theorem is referenced by:  fsumshft  11471  fprodshft  11645
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