Theorem iseqf1olemkle 10934

Description: Lemma for seq3f1o 10954. (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 10428	. . . . . 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 9768	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. RR )
6		iseqf1olemkle.j	. . . . . . . . 9 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		f1ocnv 5652	. . . . . . . . 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 5639	. . . . . . . 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 5844	. . . . . 6 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
12		elfzelz 10428	. . . . . 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 9768	. . 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 8445	. 2 |- ( ( ph /\ K < ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
18	3	zred 9768	. . 3 |- ( ph -> K e. RR )
19		eqle 8417	. . 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 5984	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
24	21, 22, 23	syl2anc 415	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
25		fveq2 5695	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
26		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
27	25, 26	eqeq12d 2253	. . . . 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 10424	. . . . . . . 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 10557	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
36	32, 33, 34, 35	syl3anbrc 1212	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
37	27, 29, 36	rspcdva 2934	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
38	24, 37	eqtr3d 2273	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
39	38, 20	syldan 282	. 2 |- ( ( ph /\ ( `' J ` K ) < K ) -> K <_ ( `' J ` K ) )
40		ztri3or 9687	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
41	3, 13, 40	syl2anc 415	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
42	17, 20, 39, 41	mpjao3dan 1348	1 |- ( ph -> K <_ ( `' J ` K ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ w3o 1008   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4130   `'ccnv 4773   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   RRcr 8178   < clt 8360   <_ cle 8361   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411  ..^cfzo 10549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550

This theorem is used by:  iseqf1olemqk  10944  seq3f1olemqsumkj  10948  seq3f1olemqsumk  10949