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Theorem iseqf1olemkle 10392
Description: Lemma for seq3f1o 10412. (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 9934 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
31, 2syl 14 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
43adantr 274 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  ZZ )
54zred 9291 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  RR )
6 iseqf1olemkle.j . . . . . . . . 9  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7 f1ocnv 5429 . . . . . . . . 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 5416 . . . . . . . 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, 1ffvelrnd 5605 . . . . . 6  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
12 elfzelz 9934 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
1311, 12syl 14 . . . . 5  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
1413adantr 274 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  ZZ )
1514zred 9291 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  RR )
16 simpr 109 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <  ( `' J `  K ) )
175, 15, 16ltled 7998 . 2  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
183zred 9291 . . 3  |-  ( ph  ->  K  e.  RR )
19 eqle 7971 . . 3  |-  ( ( K  e.  RR  /\  K  =  ( `' J `  K )
)  ->  K  <_  ( `' J `  K ) )
2018, 19sylan 281 . 2  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
216adantr 274 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
221adantr 274 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ( M ... N
) )
23 f1ocnvfv2 5730 . . . . 5  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
2421, 22, 23syl2anc 409 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  K )
25 fveq2 5470 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  ( J `  x )  =  ( J `  ( `' J `  K ) ) )
26 id 19 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  x  =  ( `' J `  K ) )
2725, 26eqeq12d 2172 . . . . 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 274 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  A. x  e.  ( M..^ K ) ( J `  x
)  =  x )
30 elfzuz 9930 . . . . . . . 8  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
3111, 30syl 14 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  (
ZZ>= `  M ) )
3231adantr 274 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
333adantr 274 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ZZ )
34 simpr 109 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  <  K )
35 elfzo2 10058 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M..^ K
)  <->  ( ( `' J `  K )  e.  ( ZZ>= `  M
)  /\  K  e.  ZZ  /\  ( `' J `  K )  <  K
) )
3632, 33, 34, 35syl3anbrc 1166 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( M..^ K
) )
3727, 29, 36rspcdva 2821 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  ( `' J `  K ) )
3824, 37eqtr3d 2192 . . 3  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  =  ( `' J `  K ) )
3938, 20syldan 280 . 2  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  <_  ( `' J `  K ) )
40 ztri3or 9215 . . 3  |-  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> 
( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
413, 13, 40syl2anc 409 . 2  |-  ( ph  ->  ( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
4217, 20, 39, 41mpjao3dan 1289 1  |-  ( ph  ->  K  <_  ( `' J `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ w3o 962    = wceq 1335    e. wcel 2128   A.wral 2435   class class class wbr 3967   `'ccnv 4587   -->wf 5168   -1-1-onto->wf1o 5171   ` cfv 5172  (class class class)co 5826   RRcr 7733    < clt 7914    <_ cle 7915   ZZcz 9172   ZZ>=cuz 9444   ...cfz 9918  ..^cfzo 10050
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-addcom 7834  ax-addass 7836  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-ltadd 7850
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-fz 9919  df-fzo 10051
This theorem is referenced by:  iseqf1olemqk  10402  seq3f1olemqsumkj  10406  seq3f1olemqsumk  10407
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