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Theorem iseqf1olemqcl 10483
Description: Lemma for seq3f1o 10501. (Contributed by Jim Kingdon, 27-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqcl.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemqcl.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqcl.a  |-  ( ph  ->  A  e.  ( M ... N ) )
Assertion
Ref Expression
iseqf1olemqcl  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  e.  ( M ... N ) )

Proof of Theorem iseqf1olemqcl
StepHypRef Expression
1 iseqf1olemqcl.k . . . 4  |-  ( ph  ->  K  e.  ( M ... N ) )
21ad2antrr 488 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  A  =  K )  ->  K  e.  ( M ... N
) )
3 iseqf1olemqcl.j . . . . . 6  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4 f1of 5461 . . . . . 6  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  J :
( M ... N
) --> ( M ... N ) )
53, 4syl 14 . . . . 5  |-  ( ph  ->  J : ( M ... N ) --> ( M ... N ) )
65ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  J : ( M ... N ) --> ( M ... N ) )
71ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  ( M ... N ) )
8 elfzel1 10021 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
97, 8syl 14 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  e.  ZZ )
10 elfzel2 10020 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
117, 10syl 14 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  N  e.  ZZ )
12 iseqf1olemqcl.a . . . . . . . . 9  |-  ( ph  ->  A  e.  ( M ... N ) )
13 elfzelz 10022 . . . . . . . . 9  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
1412, 13syl 14 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
1514ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  ZZ )
16 peano2zm 9289 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  ZZ )
1715, 16syl 14 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ZZ )
189, 11, 173jca 1177 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1
)  e.  ZZ ) )
199zred 9373 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  e.  RR )
20 elfzelz 10022 . . . . . . . . 9  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
217, 20syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  ZZ )
2221zred 9373 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  RR )
2317zred 9373 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
24 elfzle1 10024 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  M  <_  K )
257, 24syl 14 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  <_  K )
26 simpr 110 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  -.  A  =  K )
27 eqcom 2179 . . . . . . . . . 10  |-  ( A  =  K  <->  K  =  A )
2826, 27sylnib 676 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  -.  K  =  A )
29 elfzle1 10024 . . . . . . . . . . 11  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  K  <_  A )
3029ad2antlr 489 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <_  A )
31 zleloe 9298 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
3221, 15, 31syl2anc 411 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
3330, 32mpbid 147 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  \/  K  =  A
) )
3428, 33ecased 1349 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <  A )
35 zltlem1 9308 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
3621, 15, 35syl2anc 411 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
3734, 36mpbid 147 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <_  ( A  -  1 ) )
3819, 22, 23, 25, 37letrd 8079 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  <_  ( A  -  1 ) )
3915zred 9373 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
4011zred 9373 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  N  e.  RR )
4139lem1d 8888 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
4212ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  ( M ... N ) )
43 elfzle2 10025 . . . . . . . 8  |-  ( A  e.  ( M ... N )  ->  A  <_  N )
4442, 43syl 14 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  <_  N )
4523, 39, 40, 41, 44letrd 8079 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  N )
4638, 45jca 306 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( M  <_  ( A  -  1 )  /\  ( A  - 
1 )  <_  N
) )
47 elfz2 10013 . . . . 5  |-  ( ( A  -  1 )  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1 )  e.  ZZ )  /\  ( M  <_  ( A  - 
1 )  /\  ( A  -  1 )  <_  N ) ) )
4818, 46, 47sylanbrc 417 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( M ... N ) )
496, 48ffvelcdmd 5652 . . 3  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( J `  ( A  -  1 ) )  e.  ( M ... N ) )
501, 20syl 14 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
51 zdceq 9326 . . . . 5  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
5214, 50, 51syl2anc 411 . . . 4  |-  ( ph  -> DECID  A  =  K )
5352adantr 276 . . 3  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  -> DECID  A  =  K
)
542, 49, 53ifcldadc 3563 . 2  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  if ( A  =  K ,  K ,  ( J `
 ( A  - 
1 ) ) )  e.  ( M ... N ) )
555, 12ffvelcdmd 5652 . . 3  |-  ( ph  ->  ( J `  A
)  e.  ( M ... N ) )
5655adantr 276 . 2  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( J `  A )  e.  ( M ... N
) )
57 f1ocnv 5474 . . . . . 6  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
58 f1of 5461 . . . . . 6  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
593, 57, 583syl 17 . . . . 5  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
6059, 1ffvelcdmd 5652 . . . 4  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
61 elfzelz 10022 . . . 4  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
6260, 61syl 14 . . 3  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
63 fzdcel 10037 . . 3  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
6414, 50, 62, 63syl3anc 1238 . 2  |-  ( ph  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
6554, 56, 64ifcldadc 3563 1  |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `  A
) )  e.  ( M ... N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   ifcif 3534   class class class wbr 4003   `'ccnv 4625   -->wf 5212   -1-1-onto->wf1o 5215   ` cfv 5216  (class class class)co 5874   1c1 7811    < clt 7990    <_ cle 7991    - cmin 8126   ZZcz 9251   ...cfz 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007
This theorem is referenced by:  iseqf1olemqval  10484  iseqf1olemqf  10488
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