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Theorem mptfzshft 13572
Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 13574. (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 6320 . . . 4  |-  ( j  -  K )  e. 
_V
2 eqid 2467 . . . 4  |-  ( j  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( j  -  K ) )  =  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )
31, 2fnmpti 5715 . . 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 6320 . . . 4  |-  ( k  +  K )  e. 
_V
6 eqid 2467 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( k  +  K ) )  =  ( k  e.  ( M ... N
)  |->  ( k  +  K ) )
75, 6fnmpti 5715 . . 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 6310 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( k  +  K
)  =  ( ( j  -  K )  +  K ) )
10 elfzelz 11700 . . . . . . . . . . . 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 10881 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  CC )
15 zcn 10881 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
16 npcan 9841 . . . . . . . . . . . 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 2509 . . . . . . . . 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 2556 . . . . . . . 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 10983 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
( j  -  K
)  e.  ZZ )
278, 26eqeltrd 2555 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) ) )  -> 
k  e.  ZZ )
28 fzaddel 11730 . . . . . . . . 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 1229 . . . . . . . 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 11700 . . . . . . . . . . 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 1229 . . . . . . . . 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 2555 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
j  e.  ( ( M  +  K ) ... ( N  +  K ) ) )
4232oveq1d 6310 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) )  -> 
( j  -  K
)  =  ( ( k  +  K )  -  K ) )
43 zcn 10881 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  CC )
44 pncan 9838 . . . . . . . . . 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 2509 . . . . . . 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 830 . . . . 5  |-  ( ph  ->  ( ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  /\  k  =  ( j  -  K
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( k  +  K
) ) ) )
5049mptcnv 5414 . . . 4  |-  ( ph  ->  `' ( j  e.  ( ( M  +  K ) ... ( N  +  K )
)  |->  ( j  -  K ) )  =  ( k  e.  ( M ... N ) 
|->  ( k  +  K
) ) )
5150fneq1d 5677 . . 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 5830 . 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 1379    e. wcel 1767    |-> cmpt 4511   `'ccnv 5004    Fn wfn 5589   -1-1-onto->wf1o 5593  (class class class)co 6295   CCcc 9502    + caddc 9507    - cmin 9817   ZZcz 10876   ...cfz 11684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685
This theorem is referenced by:  fsumshft  13574  gsummptshft  16827  fprodshft  29040
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