Theorem iseqf1olemkle 10644

Description: Lemma for seq3f1o 10664. (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 10149	. . . . . 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 9497	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. RR )
6		iseqf1olemkle.j	. . . . . . . . 9 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		f1ocnv 5537	. . . . . . . . 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 5524	. . . . . . . 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 5718	. . . . . 6 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
12		elfzelz 10149	. . . . . 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 9497	. . 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 8193	. 2 |- ( ( ph /\ K < ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
18	3	zred 9497	. . 3 |- ( ph -> K e. RR )
19		eqle 8166	. . 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 5849	. . . . 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 5578	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
26		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
27	25, 26	eqeq12d 2220	. . . . 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 10145	. . . . . . . 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 10274	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
36	32, 33, 34, 35	syl3anbrc 1184	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
37	27, 29, 36	rspcdva 2882	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
38	24, 37	eqtr3d 2240	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
39	38, 20	syldan 282	. 2 |- ( ( ph /\ ( `' J ` K ) < K ) -> K <_ ( `' J ` K ) )
40		ztri3or 9417	. . 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 1320	1 |- ( ph -> K <_ ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ w3o 980   = wceq 1373   e. wcel 2176   A.wral 2484   class class class wbr 4045   `'ccnv 4675   -->wf 5268   -1-1-onto->wf1o 5271   ` cfv 5272  (class class class)co 5946   RRcr 7926   < clt 8109   <_ cle 8110   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132  ..^cfzo 10266

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-fzo 10267

This theorem is referenced by:  iseqf1olemqk  10654  seq3f1olemqsumkj  10658  seq3f1olemqsumk  10659