Theorem iseqf1olemkle 10719

Description: Lemma for seq3f1o 10739. (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

Step	Hyp	Ref	Expression
1		iseqf1olemkle.k	. . . . . 6 |- ( ph -> K e. ( M ... N ) )
2		elfzelz 10221	. . . . . 6 |- ( K e. ( M ... N ) -> K e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> K e. ZZ )
4	3	adantr 276	. . . 4 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. ZZ )
5	4	zred 9569	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. RR )
6		iseqf1olemkle.j	. . . . . . . . 9 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		f1ocnv 5585	. . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
8	6, 7	syl 14	. . . . . . . 8 |- ( ph -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
9		f1of 5572	. . . . . . . 8 |- ( `' J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) --> ( M ... N ) )
10	8, 9	syl 14	. . . . . . 7 |- ( ph -> `' J : ( M ... N ) --> ( M ... N ) )
11	10, 1	ffvelcdmd 5771	. . . . . 6 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
12		elfzelz 10221	. . . . . 6 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
13	11, 12	syl 14	. . . . 5 |- ( ph -> ( `' J ` K ) e. ZZ )
14	13	adantr 276	. . . 4 |- ( ( ph /\ K < ( `' J ` K ) ) -> ( `' J ` K ) e. ZZ )
15	14	zred 9569	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> ( `' J ` K ) e. RR )
16		simpr 110	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K < ( `' J ` K ) )
17	5, 15, 16	ltled 8265	. 2 |- ( ( ph /\ K < ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
18	3	zred 9569	. . 3 |- ( ph -> K e. RR )
19		eqle 8238	. . 3 |- ( ( K e. RR /\ K = ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
20	18, 19	sylan 283	. 2 |- ( ( ph /\ K = ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
21	6	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
22	1	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ( M ... N ) )
23		f1ocnvfv2 5902	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
24	21, 22, 23	syl2anc 411	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
25		fveq2 5627	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
26		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
27	25, 26	eqeq12d 2244	. . . . 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 )
29	28	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> A. x e. ( M..^ K ) ( J ` x ) = x )
30		elfzuz 10217	. . . . . . . 8 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
31	11, 30	syl 14	. . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( ZZ>= ` M ) )
32	31	adantr 276	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
33	3	adantr 276	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ZZ )
34		simpr 110	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) < K )
35		elfzo2 10346	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
36	32, 33, 34, 35	syl3anbrc 1205	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
37	27, 29, 36	rspcdva 2912	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
38	24, 37	eqtr3d 2264	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
39	38, 20	syldan 282	. 2 |- ( ( ph /\ ( `' J ` K ) < K ) -> K <_ ( `' J ` K ) )
40		ztri3or 9489	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
41	3, 13, 40	syl2anc 411	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
42	17, 20, 39, 41	mpjao3dan 1341	1 |- ( ph -> K <_ ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ w3o 1001   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4083   `'ccnv 4718   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   RRcr 7998   < clt 8181   <_ cle 8182   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204  ..^cfzo 10338

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-fzo 10339

This theorem is referenced by:  iseqf1olemqk  10729  seq3f1olemqsumkj  10733  seq3f1olemqsumk  10734