ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqf1olemqf1o Unicode version

Theorem iseqf1olemqf1o 9976
Description: Lemma for seq3f1o 9987. 
Q is a permutation of  ( M ... N
).  Q is formed from the constant portion of  J, followed by the single element  K (at position  K), followed by the rest of J (with the  K deleted and the elements before  K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqf.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemqf.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqf.q  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
Assertion
Ref Expression
iseqf1olemqf1o  |-  ( ph  ->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Distinct variable groups:    u, J    u, K    u, M    u, N    ph, u
Allowed substitution hint:    Q( u)

Proof of Theorem iseqf1olemqf1o
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqf1olemqf.k . . . 4  |-  ( ph  ->  K  e.  ( M ... N ) )
2 iseqf1olemqf.j . . . 4  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
3 iseqf1olemqf.q . . . 4  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
41, 2, 3iseqf1olemqf 9974 . . 3  |-  ( ph  ->  Q : ( M ... N ) --> ( M ... N ) )
51ad2antrr 473 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  K  e.  ( M ... N ) )
62ad2antrr 473 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) )
7 simplrl 503 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  v  e.  ( M ... N ) )
8 simplrr 504 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  w  e.  ( M ... N ) )
9 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  ( Q `  v )  =  ( Q `  w ) )
105, 6, 3, 7, 8, 9iseqf1olemmo 9975 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  v  =  w )
1110ex 114 . . . 4  |-  ( (
ph  /\  ( v  e.  ( M ... N
)  /\  w  e.  ( M ... N ) ) )  ->  (
( Q `  v
)  =  ( Q `
 w )  -> 
v  =  w ) )
1211ralrimivva 2456 . . 3  |-  ( ph  ->  A. v  e.  ( M ... N ) A. w  e.  ( M ... N ) ( ( Q `  v )  =  ( Q `  w )  ->  v  =  w ) )
13 dff13 5561 . . 3  |-  ( Q : ( M ... N ) -1-1-> ( M ... N )  <->  ( Q : ( M ... N ) --> ( M ... N )  /\  A. v  e.  ( M ... N ) A. w  e.  ( M ... N ) ( ( Q `  v )  =  ( Q `  w )  ->  v  =  w ) ) )
144, 12, 13sylanbrc 409 . 2  |-  ( ph  ->  Q : ( M ... N ) -1-1-> ( M ... N ) )
15 elfzel1 9493 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
161, 15syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
17 elfzel2 9492 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
181, 17syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
1916, 18fzfigd 9892 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
20 enrefg 6535 . . . 4  |-  ( ( M ... N )  e.  Fin  ->  ( M ... N )  ~~  ( M ... N ) )
2119, 20syl 14 . . 3  |-  ( ph  ->  ( M ... N
)  ~~  ( M ... N ) )
22 f1finf1o 6710 . . 3  |-  ( ( ( M ... N
)  ~~  ( M ... N )  /\  ( M ... N )  e. 
Fin )  ->  ( Q : ( M ... N ) -1-1-> ( M ... N )  <->  Q :
( M ... N
)
-1-1-onto-> ( M ... N ) ) )
2321, 19, 22syl2anc 404 . 2  |-  ( ph  ->  ( Q : ( M ... N )
-1-1-> ( M ... N
)  <->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) ) )
2414, 23mpbid 146 1  |-  ( ph  ->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   A.wral 2360   ifcif 3397   class class class wbr 3851    |-> cmpt 3905   `'ccnv 4450   -->wf 5024   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028  (class class class)co 5666    ~~ cen 6509   Fincfn 6511   1c1 7405    - cmin 7707   ZZcz 8804   ...cfz 9478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-1o 6195  df-er 6306  df-en 6512  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479
This theorem is referenced by:  seq3f1olemqsumkj  9981  seq3f1olemqsumk  9982  seq3f1olemstep  9984
  Copyright terms: Public domain W3C validator