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Theorem mptfzshft 13245
Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 13247. (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 ovex 6116 . . . 4  |-  ( j  -  K )  e. 
_V
2 eqid 2443 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
31, 2fnmpti 5539 . . 3  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K
) )
43a1i 11 . 2  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) ) )
5 ovex 6116 . . . 4  |-  ( k  +  K )  e. 
_V
6 eqid 2443 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
75, 6fnmpti 5539 . . 3  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  Fn  ( M ... N )
8 simprr 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  =  ( j  -  K ) )
98oveq1d 6106 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
10 elfzelz 11453 . . . . . . . . . . . 12  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  j  e.  ZZ )
1110ad2antrl 727 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ZZ )
12 mptfzshft.1 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ZZ )
1312adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  K  e.  ZZ )
14 zcn 10651 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  CC )
15 zcn 10651 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
16 npcan 9619 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  K  e.  CC )  ->  ( ( j  -  K )  +  K
)  =  j )
1714, 15, 16syl2an 477 . . . . . . . . . . 11  |-  ( ( j  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( j  -  K )  +  K
)  =  j )
1811, 13, 17syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( ( j  -  K )  +  K
)  =  j )
199, 18eqtr2d 2476 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  =  ( k  +  K ) )
20 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
2119, 20eqeltrrd 2518 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
22 mptfzshft.2 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ZZ )
2322adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  M  e.  ZZ )
24 mptfzshft.3 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
2524adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  ->  N  e.  ZZ )
2611, 13zsubcld 10752 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
278, 26eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
28 fzaddel 11493 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( k  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
2923, 25, 27, 13, 28syl22anc 1219 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3021, 29mpbird 232 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ( M ... N ) )
3130, 19jca 532 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( k  +  K ) ) )
32 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  =  ( k  +  K ) )
33 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ( M ... N ) )
3422adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  M  e.  ZZ )
3524adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  N  e.  ZZ )
36 elfzelz 11453 . . . . . . . . . . 11  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3736ad2antrl 727 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  e.  ZZ )
3812adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  ->  K  e.  ZZ )
3934, 35, 37, 38, 28syl22anc 1219 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
4033, 39mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( k  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4132, 40eqeltrd 2517 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4232oveq1d 6106 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
43 zcn 10651 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
44 pncan 9616 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  +  K )  -  K
)  =  k )
4543, 15, 44syl2an 477 . . . . . . . . 9  |-  ( ( k  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( k  +  K )  -  K
)  =  k )
4637, 38, 45syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( ( k  +  K )  -  K
)  =  k )
4742, 46eqtr2d 2476 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
k  =  ( j  -  K ) )
4841, 47jca 532 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  /\  k  =  ( j  -  K ) ) )
4931, 48impbida 828 . . . . 5  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5049mptcnv 5239 . . . 4  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
5150fneq1d 5501 . . 3  |-  ( ph  ->  ( `' ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )  Fn  ( M ... N
) ) )
527, 51mpbiri 233 . 2  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) )
53 dff1o4 5649 . 2  |-  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N )  <->  ( (
j  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  ( j  -  K ) )  Fn  ( ( M  +  K ) ... ( N  +  K ) )  /\  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  Fn  ( M ... N
) ) )
544, 52, 53sylanbrc 664 1  |-  ( ph  ->  ( j  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( j  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4350   `'ccnv 4839    Fn wfn 5413   -1-1-onto->wf1o 5417  (class class class)co 6091   CCcc 9280    + caddc 9285    - cmin 9595   ZZcz 10646   ...cfz 11437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438
This theorem is referenced by:  fsumshft  13247  gsummptshft  16429
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