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Theorem iseqf1olemqcl 10214
Description: Lemma for seq3f1o 10232. (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 479 . . 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 5335 . . . . . 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 479 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  J : ( M ... N ) --> ( M ... N ) )
71ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  ( M ... N ) )
8 elfzel1 9760 . . . . . . 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 9759 . . . . . . 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 9761 . . . . . . . . 9  |-  ( A  e.  ( M ... N )  ->  A  e.  ZZ )
1412, 13syl 14 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
1514ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  ZZ )
16 peano2zm 9050 . . . . . . 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 1146 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( A  -  1
)  e.  ZZ ) )
199zred 9131 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  e.  RR )
20 elfzelz 9761 . . . . . . . . 9  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
217, 20syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  ZZ )
2221zred 9131 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  e.  RR )
2317zred 9131 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  RR )
24 elfzle1 9762 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  M  <_  K )
257, 24syl 14 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  <_  K )
26 simpr 109 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  -.  A  =  K )
27 eqcom 2119 . . . . . . . . . 10  |-  ( A  =  K  <->  K  =  A )
2826, 27sylnib 650 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  -.  K  =  A )
29 elfzle1 9762 . . . . . . . . . . 11  |-  ( A  e.  ( K ... ( `' J `  K ) )  ->  K  <_  A )
3029ad2antlr 480 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <_  A )
31 zleloe 9059 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
3221, 15, 31syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <_  A  <->  ( K  <  A  \/  K  =  A )
) )
3330, 32mpbid 146 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  \/  K  =  A
) )
3428, 33ecased 1312 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <  A )
35 zltlem1 9069 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
3621, 15, 35syl2anc 408 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( K  <  A  <->  K  <_  ( A  - 
1 ) ) )
3734, 36mpbid 146 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  K  <_  ( A  -  1 ) )
3819, 22, 23, 25, 37letrd 7854 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  M  <_  ( A  -  1 ) )
3915zred 9131 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  RR )
4011zred 9131 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  N  e.  RR )
4139lem1d 8655 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  A )
4212ad2antrr 479 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  A  e.  ( M ... N ) )
43 elfzle2 9763 . . . . . . . 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 7854 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  <_  N )
4638, 45jca 304 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( M  <_  ( A  -  1 )  /\  ( A  - 
1 )  <_  N
) )
47 elfz2 9752 . . . . 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 413 . . . 4  |-  ( ( ( ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  /\  -.  A  =  K )  ->  ( A  -  1 )  e.  ( M ... N ) )
496, 48ffvelrnd 5524 . . 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 9084 . . . . 5  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ )  -> DECID  A  =  K )
5214, 50, 51syl2anc 408 . . . 4  |-  ( ph  -> DECID  A  =  K )
5352adantr 274 . . 3  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  -> DECID  A  =  K
)
542, 49, 53ifcldadc 3471 . 2  |-  ( (
ph  /\  A  e.  ( K ... ( `' J `  K ) ) )  ->  if ( A  =  K ,  K ,  ( J `
 ( A  - 
1 ) ) )  e.  ( M ... N ) )
555, 12ffvelrnd 5524 . . 3  |-  ( ph  ->  ( J `  A
)  e.  ( M ... N ) )
5655adantr 274 . 2  |-  ( (
ph  /\  -.  A  e.  ( K ... ( `' J `  K ) ) )  ->  ( J `  A )  e.  ( M ... N
) )
57 f1ocnv 5348 . . . . . 6  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
58 f1of 5335 . . . . . 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, 1ffvelrnd 5524 . . . 4  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
61 elfzelz 9761 . . . 4  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
6260, 61syl 14 . . 3  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
63 fzdcel 9775 . . 3  |-  ( ( A  e.  ZZ  /\  K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
6414, 50, 62, 63syl3anc 1201 . 2  |-  ( ph  -> DECID  A  e.  ( K ... ( `' J `  K ) ) )
6554, 56, 64ifcldadc 3471 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 103    <-> wb 104    \/ wo 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465   ifcif 3444   class class class wbr 3899   `'ccnv 4508   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742   1c1 7589    < clt 7768    <_ cle 7769    - cmin 7901   ZZcz 9012   ...cfz 9745
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746
This theorem is referenced by:  iseqf1olemqval  10215  iseqf1olemqf  10219
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