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Theorem iseqf1olemqf1o 10436
Description: Lemma for seq3f1o 10447. 
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 10434 . . 3  |-  ( ph  ->  Q : ( M ... N ) --> ( M ... N ) )
51ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  K  e.  ( M ... N ) )
62ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) )
7 simplrl 530 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  v  e.  ( M ... N ) )
8 simplrr 531 . . . . . 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 10435 . . . . 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 2552 . . 3  |-  ( ph  ->  A. v  e.  ( M ... N ) A. w  e.  ( M ... N ) ( ( Q `  v )  =  ( Q `  w )  ->  v  =  w ) )
13 dff13 5744 . . 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 415 . 2  |-  ( ph  ->  Q : ( M ... N ) -1-1-> ( M ... N ) )
15 elfzel1 9967 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
161, 15syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
17 elfzel2 9966 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
181, 17syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
1916, 18fzfigd 10374 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
20 enrefg 6738 . . . 4  |-  ( ( M ... N )  e.  Fin  ->  ( M ... N )  ~~  ( M ... N ) )
2119, 20syl 14 . . 3  |-  ( ph  ->  ( M ... N
)  ~~  ( M ... N ) )
22 f1finf1o 6920 . . 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 409 . 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 1348    e. wcel 2141   A.wral 2448   ifcif 3525   class class class wbr 3987    |-> cmpt 4048   `'ccnv 4608   -->wf 5192   -1-1->wf1 5193   -1-1-onto->wf1o 5195   ` cfv 5196  (class class class)co 5850    ~~ cen 6712   Fincfn 6714   1c1 7762    - cmin 8077   ZZcz 9199   ...cfz 9952
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-er 6509  df-en 6715  df-fin 6717  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953
This theorem is referenced by:  seq3f1olemqsumkj  10441  seq3f1olemqsumk  10442  seq3f1olemstep  10444
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