Theorem iseqf1olemklt 10258

Description: Lemma for seq3f1o 10277. (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

Step	Hyp	Ref	Expression
1		iseqf1olemklt.kj	. . 3 |- ( ph -> K =/= ( `' J ` K ) )
2	1	neneqd 2329	. 2 |- ( ph -> -. K = ( `' J ` K ) )
3		iseqf1olemklt.j	. . . . . 6 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4	3	adantr 274	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
5		iseqf1olemklt.k	. . . . . 6 |- ( ph -> K e. ( M ... N ) )
6	5	adantr 274	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ( M ... N ) )
7		f1ocnvfv2 5679	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
8	4, 6, 7	syl2anc 408	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
9		fveq2 5421	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
10		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
11	9, 10	eqeq12d 2154	. . . . 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 )
13	12	adantr 274	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> A. x e. ( M..^ K ) ( J ` x ) = x )
14		f1ocnv 5380	. . . . . . . . . . 11 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( ph -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
16		f1of 5367	. . . . . . . . . 10 |- ( `' J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) --> ( M ... N ) )
17	15, 16	syl 14	. . . . . . . . 9 |- ( ph -> `' J : ( M ... N ) --> ( M ... N ) )
18	17, 5	ffvelrnd 5556	. . . . . . . 8 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
19		elfzuz 9802	. . . . . . . 8 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
20	18, 19	syl 14	. . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( ZZ>= ` M ) )
21	20	adantr 274	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
22		elfzelz 9806	. . . . . . . 8 |- ( K e. ( M ... N ) -> K e. ZZ )
23	5, 22	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
24	23	adantr 274	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ZZ )
25		simpr 109	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) < K )
26		elfzo2 9927	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
27	21, 24, 25, 26	syl3anbrc 1165	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
28	11, 13, 27	rspcdva 2794	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
29	8, 28	eqtr3d 2174	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
30	2, 29	mtand 654	. 2 |- ( ph -> -. ( `' J ` K ) < K )
31		elfzelz 9806	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
32	18, 31	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
33		ztri3or 9097	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
34	23, 32, 33	syl2anc 408	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
35	2, 30, 34	ecase23d 1328	1 |- ( ph -> K < ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ w3o 961   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   class class class wbr 3929   `'ccnv 4538   -->wf 5119   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   < clt 7800   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790  ..^cfzo 9919

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791  df-fzo 9920

This theorem is referenced by:  seq3f1olemqsumkj  10271