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Theorem mptfzshft 11242
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 2140 . 2  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
2 elfzelz 9836 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
32adantl 275 . . 3  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  j  e.  ZZ )
4 mptfzshft.1 . . . 4  |-  ( ph  ->  K  e.  ZZ )
54adantr 274 . . 3  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  K  e.  ZZ )
63, 5zsubcld 9201 . 2  |-  ( (
ph  /\  j  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
j  -  K )  e.  ZZ )
7 elfzelz 9836 . . . 4  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
87adantl 275 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  k  e.  ZZ )
94adantr 274 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  K  e.  ZZ )
108, 9zaddcld 9200 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( k  +  K )  e.  ZZ )
11 simprr 522 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
1211oveq1d 5796 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
132ad2antrl 482 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
144adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
15 zcn 9082 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  j  e.  CC )
16 zcn 9082 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
17 npcan 7994 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  K  e.  CC )  ->  ( ( j  -  K )  +  K
)  =  j )
1815, 16, 17syl2an 287 . . . . . . . 8  |-  ( ( j  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( j  -  K )  +  K
)  =  j )
1913, 14, 18syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
2012, 19eqtr2d 2174 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
21 simprl 521 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2220, 21eqeltrrd 2218 . . . . 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 274 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
25 mptfzshft.3 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2625adantr 274 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
2713, 14zsubcld 9201 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
2811, 27eqeltrd 2217 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
29 fzaddel 9869 . . . . . 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 1218 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3122, 30mpbird 166 . . . 4  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ( M ... N ) )
3231, 20jca 304 . . 3  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
33 simprr 522 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
34 simprl 521 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3523adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3625adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
377ad2antrl 482 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
384adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
3935, 36, 37, 38, 29syl22anc 1218 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
4034, 39mpbid 146 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4133, 40eqeltrd 2217 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4233oveq1d 5796 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
43 zcn 9082 . . . . . . 7  |-  ( k  e.  ZZ  ->  k  e.  CC )
44 pncan 7991 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  +  K )  -  K
)  =  k )
4543, 16, 44syl2an 287 . . . . . 6  |-  ( ( k  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( k  +  K )  -  K
)  =  k )
4637, 38, 45syl2anc 409 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
4742, 46eqtr2d 2174 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
4841, 47jca 304 . . 3  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
4932, 48impbida 586 . 2  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
501, 6, 10, 49f1od 5980 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 103    <-> wb 104    = wceq 1332    e. wcel 1481    |-> cmpt 3996   -1-1-onto->wf1o 5129  (class class class)co 5781   CCcc 7641    + caddc 7646    - cmin 7956   ZZcz 9077   ...cfz 9820
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-fz 9821
This theorem is referenced by:  fsumshft  11244
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