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Theorem iseqf1olemklt 10516
Description: Lemma for seq3f1o 10535. (Contributed by Jim Kingdon, 21-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemklt.n  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqf1olemklt.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemklt.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemklt.const  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
iseqf1olemklt.kj  |-  ( ph  ->  K  =/=  ( `' J `  K ) )
Assertion
Ref Expression
iseqf1olemklt  |-  ( ph  ->  K  <  ( `' J `  K ) )
Distinct variable groups:    x, J    x, K    x, M
Allowed substitution hints:    ph( x)    N( x)

Proof of Theorem iseqf1olemklt
StepHypRef Expression
1 iseqf1olemklt.kj . . 3  |-  ( ph  ->  K  =/=  ( `' J `  K ) )
21neneqd 2381 . 2  |-  ( ph  ->  -.  K  =  ( `' J `  K ) )
3 iseqf1olemklt.j . . . . . 6  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
43adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  J : ( M ... N ) -1-1-onto-> ( M ... N
) )
5 iseqf1olemklt.k . . . . . 6  |-  ( ph  ->  K  e.  ( M ... N ) )
65adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ( M ... N
) )
7 f1ocnvfv2 5800 . . . . 5  |-  ( ( J : ( M ... N ) -1-1-onto-> ( M ... N )  /\  K  e.  ( M ... N ) )  -> 
( J `  ( `' J `  K ) )  =  K )
84, 6, 7syl2anc 411 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  K )
9 fveq2 5534 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  ( J `  x )  =  ( J `  ( `' J `  K ) ) )
10 id 19 . . . . . 6  |-  ( x  =  ( `' J `  K )  ->  x  =  ( `' J `  K ) )
119, 10eqeq12d 2204 . . . . 5  |-  ( x  =  ( `' J `  K )  ->  (
( J `  x
)  =  x  <->  ( J `  ( `' J `  K ) )  =  ( `' J `  K ) ) )
12 iseqf1olemklt.const . . . . . 6  |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `  x )  =  x )
1312adantr 276 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  A. x  e.  ( M..^ K ) ( J `  x
)  =  x )
14 f1ocnv 5493 . . . . . . . . . . 11  |-  ( J : ( M ... N ) -1-1-onto-> ( M ... N
)  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N
) )
153, 14syl 14 . . . . . . . . . 10  |-  ( ph  ->  `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
16 f1of 5480 . . . . . . . . . 10  |-  ( `' J : ( M ... N ) -1-1-onto-> ( M ... N )  ->  `' J : ( M ... N ) --> ( M ... N ) )
1715, 16syl 14 . . . . . . . . 9  |-  ( ph  ->  `' J : ( M ... N ) --> ( M ... N ) )
1817, 5ffvelcdmd 5673 . . . . . . . 8  |-  ( ph  ->  ( `' J `  K )  e.  ( M ... N ) )
19 elfzuz 10051 . . . . . . . 8  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
2018, 19syl 14 . . . . . . 7  |-  ( ph  ->  ( `' J `  K )  e.  (
ZZ>= `  M ) )
2120adantr 276 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( ZZ>= `  M
) )
22 elfzelz 10055 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
235, 22syl 14 . . . . . . 7  |-  ( ph  ->  K  e.  ZZ )
2423adantr 276 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  e.  ZZ )
25 simpr 110 . . . . . 6  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  <  K )
26 elfzo2 10180 . . . . . 6  |-  ( ( `' J `  K )  e.  ( M..^ K
)  <->  ( ( `' J `  K )  e.  ( ZZ>= `  M
)  /\  K  e.  ZZ  /\  ( `' J `  K )  <  K
) )
2721, 24, 25, 26syl3anbrc 1183 . . . . 5  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( `' J `  K )  e.  ( M..^ K
) )
2811, 13, 27rspcdva 2861 . . . 4  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  ( J `  ( `' J `  K )
)  =  ( `' J `  K ) )
298, 28eqtr3d 2224 . . 3  |-  ( (
ph  /\  ( `' J `  K )  <  K )  ->  K  =  ( `' J `  K ) )
302, 29mtand 666 . 2  |-  ( ph  ->  -.  ( `' J `  K )  <  K
)
31 elfzelz 10055 . . . 4  |-  ( ( `' J `  K )  e.  ( M ... N )  ->  ( `' J `  K )  e.  ZZ )
3218, 31syl 14 . . 3  |-  ( ph  ->  ( `' J `  K )  e.  ZZ )
33 ztri3or 9326 . . 3  |-  ( ( K  e.  ZZ  /\  ( `' J `  K )  e.  ZZ )  -> 
( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
3423, 32, 33syl2anc 411 . 2  |-  ( ph  ->  ( K  <  ( `' J `  K )  \/  K  =  ( `' J `  K )  \/  ( `' J `  K )  <  K
) )
352, 30, 34ecase23d 1361 1  |-  ( ph  ->  K  <  ( `' J `  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ w3o 979    = wceq 1364    e. wcel 2160    =/= wne 2360   A.wral 2468   class class class wbr 4018   `'ccnv 4643   -->wf 5231   -1-1-onto->wf1o 5234   ` cfv 5235  (class class class)co 5896    < clt 8022   ZZcz 9283   ZZ>=cuz 9558   ...cfz 10038  ..^cfzo 10172
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-fz 10039  df-fzo 10173
This theorem is referenced by:  seq3f1olemqsumkj  10529
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