Theorem iseqf1olemkle 10679

Description: Lemma for seq3f1o 10699. (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 10182	. . . . . 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 9530	. . 3 |- ( ( ph /\ K < ( `' J ` K ) ) -> K e. RR )
6		iseqf1olemkle.j	. . . . . . . . 9 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
7		f1ocnv 5557	. . . . . . . . 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 5544	. . . . . . . 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 5739	. . . . . 6 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
12		elfzelz 10182	. . . . . 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 9530	. . 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 8226	. 2 |- ( ( ph /\ K < ( `' J ` K ) ) -> K <_ ( `' J ` K ) )
18	3	zred 9530	. . 3 |- ( ph -> K e. RR )
19		eqle 8199	. . 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 5870	. . . . 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 5599	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
26		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
27	25, 26	eqeq12d 2222	. . . . 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 10178	. . . . . . . 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 10307	. . . . . 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 2889	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
38	24, 37	eqtr3d 2242	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
39	38, 20	syldan 282	. 2 |- ( ( ph /\ ( `' J ` K ) < K ) -> K <_ ( `' J ` K ) )
40		ztri3or 9450	. . 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 2178   A.wral 2486   class class class wbr 4059   `'ccnv 4692   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   RRcr 7959   < clt 8142   <_ cle 8143   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165  ..^cfzo 10299

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300

This theorem is referenced by:  iseqf1olemqk  10689  seq3f1olemqsumkj  10693  seq3f1olemqsumk  10694