Theorem iseqf1olemkle 10144

Description: Lemma for seq3f1o 10164. (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 9693	. . . . . 6 |- ( K e. ( M ... N ) -> K e. ZZ )
3	1, 2	syl 14	. . . . 5 |- ( ph -> K e. ZZ )
4	3	adantr 272	. . . 4 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. ZZ )
5	4	zred 9071	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. RR )
6		iseqf1olemkle.j	. . . . . . . . 9 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		f1ocnv 5334	. . . . . . . . 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 5321	. . . . . . . 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	ffvelrnd 5508	. . . . . 6 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
12		elfzelz 9693	. . . . . 6 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
13	11, 12	syl 14	. . . . 5 |- ( ph -> ( `' J ` K ) e. ZZ )
14	13	adantr 272	. . . 4 |- ( ( ph /\ K < ( `' J ` K ) ) -> ( `' J ` K ) e. ZZ )
15	14	zred 9071	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> ( `' J ` K ) e. RR )
16		simpr 109	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K < ( `' J ` K ) )
17	5, 15, 16	ltled 7798	. 2 |- ( ( ph /\ K < ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
18	3	zred 9071	. . 3 |- ( ph -> K e. RR )
19		eqle 7772	. . 3 |- ( ( K e. RR /\ K = ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
20	18, 19	sylan 279	. 2 |- ( ( ph /\ K = ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
21	6	adantr 272	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
22	1	adantr 272	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ( M ... N ) )
23		f1ocnvfv2 5631	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
24	21, 22, 23	syl2anc 406	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
25		fveq2 5373	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
26		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
27	25, 26	eqeq12d 2127	. . . . 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 272	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> A. x e. ( M..^ K ) ( J ` x ) = x )
30		elfzuz 9689	. . . . . . . 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 272	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
33	3	adantr 272	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ZZ )
34		simpr 109	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) < K )
35		elfzo2 9814	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
36	32, 33, 34, 35	syl3anbrc 1146	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
37	27, 29, 36	rspcdva 2763	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
38	24, 37	eqtr3d 2147	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
39	38, 20	syldan 278	. 2 |- ( ( ph /\ ( `' J ` K ) < K ) -> K <_ ( `' J ` K ) )
40		ztri3or 8995	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
41	3, 13, 40	syl2anc 406	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
42	17, 20, 39, 41	mpjao3dan 1266	1 |- ( ph -> K <_ ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ w3o 942   = wceq 1312   e. wcel 1461   A.wral 2388   class class class wbr 3893   `'ccnv 4496   -->wf 5075   -1-1-onto->wf1o 5078   ` cfv 5079  (class class class)co 5726   RRcr 7540   < clt 7718   <_ cle 7719   ZZcz 8952   ZZ>=cuz 9222   ...cfz 9677  ..^cfzo 9806

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655

This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953  df-uz 9223  df-fz 9678  df-fzo 9807

This theorem is referenced by:  iseqf1olemqk  10154  seq3f1olemqsumkj  10158  seq3f1olemqsumk  10159