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Theorem iseqf1olemkle 10501
Description: Lemma for seq3f1o 10521. (Contributed by Jim Kingdon, 21-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemkle.n  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqf1olemkle.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemkle.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemkle.const  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
Assertion
Ref Expression
iseqf1olemkle  |-  ( ph  ->  K  <_  ( `' J `  K )
)
Distinct variable groups:    x, J    x, K    x, M
Allowed substitution hints:    ph( x)    N( x)

Proof of Theorem iseqf1olemkle
StepHypRef Expression
1 iseqf1olemkle.k . . . . . 6  |-  ( ph  ->  K  e.  ( M ... N ) )
2 elfzelz 10042 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
31, 2syl 14 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
43adantr 276 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  ZZ )
54zred 9392 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  RR )
6 iseqf1olemkle.j . . . . . . . . 9  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7 f1ocnv 5488 . . . . . . . . 9  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
86, 7syl 14 . . . . . . . 8  |-  ( ph  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
9 f1of 5475 . . . . . . . 8  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
108, 9syl 14 . . . . . . 7  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
1110, 1ffvelcdmd 5667 . . . . . 6  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
12 elfzelz 10042 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
1311, 12syl 14 . . . . 5  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
1413adantr 276 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  ZZ )
1514zred 9392 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  RR )
16 simpr 110 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <  ( `' J `  K ) )
175, 15, 16ltled 8093 . 2  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
183zred 9392 . . 3  |-  ( ph  ->  K  e.  RR )
19 eqle 8066 . . 3  |-  ( ( K  e.  RR  /\  K  =  ( `' J `  K )
)  ->  K  <_  ( `' J `  K ) )
2018, 19sylan 283 . 2  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
216adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
221adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ( M ... N
) )
23 f1ocnvfv2 5794 . . . . 5  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
2421, 22, 23syl2anc 411 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  K )
25 fveq2 5529 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  ( J `  x )  =  ( J `  ( `' J `  K ) ) )
26 id 19 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  x  =  ( `' J `  K ) )
2725, 26eqeq12d 2203 . . . . 5  |-  ( x  =  ( `' J `  K )  ->  (
( J `  x
)  =  x  <->  ( J `  ( `' J `  K ) )  =  ( `' J `  K ) ) )
28 iseqf1olemkle.const . . . . . 6  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
2928adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  A. x  e.  ( M..^ K ) ( J `  x
)  =  x )
30 elfzuz 10038 . . . . . . . 8  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
3111, 30syl 14 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  (
ZZ>= `  M ) )
3231adantr 276 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
333adantr 276 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ZZ )
34 simpr 110 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  <  K )
35 elfzo2 10167 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M..^ K
)  <->  ( ( `' J `  K )  e.  ( ZZ>= `  M
)  /\  K  e.  ZZ  /\  ( `' J `  K )  <  K
) )
3632, 33, 34, 35syl3anbrc 1182 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( M..^ K
) )
3727, 29, 36rspcdva 2860 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  ( `' J `  K ) )
3824, 37eqtr3d 2223 . . 3  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  =  ( `' J `  K ) )
3938, 20syldan 282 . 2  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  <_  ( `' J `  K ) )
40 ztri3or 9313 . . 3  |-  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> 
( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
413, 13, 40syl2anc 411 . 2  |-  ( ph  ->  ( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
4217, 20, 39, 41mpjao3dan 1317 1  |-  ( ph  ->  K  <_  ( `' J `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ w3o 978    = wceq 1363    e. wcel 2159   A.wral 2467   class class class wbr 4017   `'ccnv 4639   -->wf 5226   -1-1-onto->wf1o 5229   ` cfv 5230  (class class class)co 5890   RRcr 7827    < clt 8009    <_ cle 8010   ZZcz 9270   ZZ>=cuz 9545   ...cfz 10025  ..^cfzo 10159
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-ltadd 7944
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-inn 8937  df-n0 9194  df-z 9271  df-uz 9546  df-fz 10026  df-fzo 10160
This theorem is referenced by:  iseqf1olemqk  10511  seq3f1olemqsumkj  10515  seq3f1olemqsumk  10516
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