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Theorem iseqf1olemkle 9909
Description: Lemma for seq3f1o 9929. (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 9438 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
31, 2syl 14 . . . . 5  |-  ( ph  ->  K  e.  ZZ )
43adantr 270 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  ZZ )
54zred 8866 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  e.  RR )
6 iseqf1olemkle.j . . . . . . . . 9  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7 f1ocnv 5266 . . . . . . . . 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 5253 . . . . . . . 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 5435 . . . . . 6  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
12 elfzelz 9438 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
1311, 12syl 14 . . . . 5  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
1413adantr 270 . . . 4  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  ZZ )
1514zred 8866 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  ( `' J `  K )  e.  RR )
16 simpr 108 . . 3  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <  ( `' J `  K ) )
175, 15, 16ltled 7600 . 2  |-  ( (
ph  /\  K  <  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
183zred 8866 . . 3  |-  ( ph  ->  K  e.  RR )
19 eqle 7574 . . 3  |-  ( ( K  e.  RR  /\  K  =  ( `' J `  K )
)  ->  K  <_  ( `' J `  K ) )
2018, 19sylan 277 . 2  |-  ( (
ph  /\  K  =  ( `' J `  K ) )  ->  K  <_  ( `' J `  K ) )
216adantr 270 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
221adantr 270 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ( M ... N
) )
23 f1ocnvfv2 5557 . . . . 5  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
2421, 22, 23syl2anc 403 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  K )
25 fveq2 5305 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  ( J `  x )  =  ( J `  ( `' J `  K ) ) )
26 id 19 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  x  =  ( `' J `  K ) )
2725, 26eqeq12d 2102 . . . . 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 270 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  A. x  e.  ( M..^ K ) ( J `  x
)  =  x )
30 elfzuz 9434 . . . . . . . 8  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
3111, 30syl 14 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  (
ZZ>= `  M ) )
3231adantr 270 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
333adantr 270 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ZZ )
34 simpr 108 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  <  K )
35 elfzo2 9557 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M..^ K
)  <->  ( ( `' J `  K )  e.  ( ZZ>= `  M
)  /\  K  e.  ZZ  /\  ( `' J `  K )  <  K
) )
3632, 33, 34, 35syl3anbrc 1127 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( M..^ K
) )
3727, 29, 36rspcdva 2727 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  ( `' J `  K ) )
3824, 37eqtr3d 2122 . . 3  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  =  ( `' J `  K ) )
3938, 20syldan 276 . 2  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  <_  ( `' J `  K ) )
40 ztri3or 8791 . . 3  |-  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> 
( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
413, 13, 40syl2anc 403 . 2  |-  ( ph  ->  ( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
4217, 20, 39, 41mpjao3dan 1243 1  |-  ( ph  ->  K  <_  ( `' J `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ w3o 923    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3845   `'ccnv 4437   -->wf 5011   -1-1-onto->wf1o 5014   ` cfv 5015  (class class class)co 5652   RRcr 7347    < clt 7520    <_ cle 7521   ZZcz 8748   ZZ>=cuz 9017   ...cfz 9422  ..^cfzo 9549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-fz 9423  df-fzo 9550
This theorem is referenced by:  iseqf1olemqk  9919  seq3f1olemqsumkj  9923  seq3f1olemqsumk  9924
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