Theorem iseqf1olemqcl 10760

Description: Lemma for seq3f1o 10778. (Contributed by Jim Kingdon, 27-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemqcl.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemqcl.j	|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqcl.a	|- ( ph -> A e. ( M ... N ) )

Assertion

Ref	Expression
iseqf1olemqcl	|- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )

Proof of Theorem iseqf1olemqcl

Step	Hyp	Ref	Expression
1		iseqf1olemqcl.k	. . . 4 |- ( ph -> K e. ( M ... N ) )
2	1	ad2antrr 488	. . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ A = K ) -> K e. ( M ... N ) )
3		iseqf1olemqcl.j	. . . . . 6 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4		f1of 5583	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> J : ( M ... N ) --> ( M ... N ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> J : ( M ... N ) --> ( M ... N ) )
6	5	ad2antrr 488	. . . 4 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> J : ( M ... N ) --> ( M ... N ) )
7	1	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. ( M ... N ) )
8		elfzel1 10258	. . . . . . 7 |- ( K e. ( M ... N ) -> M e. ZZ )
9	7, 8	syl 14	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. ZZ )
10		elfzel2 10257	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ZZ )
11	7, 10	syl 14	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. ZZ )
12		iseqf1olemqcl.a	. . . . . . . . 9 |- ( ph -> A e. ( M ... N ) )
13		elfzelz 10259	. . . . . . . . 9 |- ( A e. ( M ... N ) -> A e. ZZ )
14	12, 13	syl 14	. . . . . . . 8 |- ( ph -> A e. ZZ )
15	14	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. ZZ )
16		peano2zm 9516	. . . . . . 7 |- ( A e. ZZ -> ( A - 1 ) e. ZZ )
17	15, 16	syl 14	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. ZZ )
18	9, 11, 17	3jca 1203	. . . . 5 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( M e. ZZ /\ N e. ZZ /\ ( A - 1 ) e. ZZ ) )
19	9	zred 9601	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. RR )
20		elfzelz 10259	. . . . . . . . 9 |- ( K e. ( M ... N ) -> K e. ZZ )
21	7, 20	syl 14	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. ZZ )
22	21	zred 9601	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. RR )
23	17	zred 9601	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
24		elfzle1 10261	. . . . . . . 8 |- ( K e. ( M ... N ) -> M <_ K )
25	7, 24	syl 14	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ K )
26		simpr 110	. . . . . . . . . 10 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. A = K )
27		eqcom 2233	. . . . . . . . . 10 |- ( A = K <-> K = A )
28	26, 27	sylnib 682	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. K = A )
29		elfzle1 10261	. . . . . . . . . . 11 |- ( A e. ( K ... ( `' J ` K ) ) -> K <_ A )
30	29	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K <_ A )
31		zleloe 9525	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
32	21, 15, 31	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
33	30, 32	mpbid 147	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( K < A \/ K = A ) )
34	28, 33	ecased 1385	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K < A )
35		zltlem1 9536	. . . . . . . . 9 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K < A <-> K <_ ( A - 1 ) ) )
36	21, 15, 35	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( K < A <-> K <_ ( A - 1 ) ) )
37	34, 36	mpbid 147	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K <_ ( A - 1 ) )
38	19, 22, 23, 25, 37	letrd 8302	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ ( A - 1 ) )
39	15	zred 9601	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. RR )
40	11	zred 9601	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. RR )
41	39	lem1d 9112	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ A )
42	12	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. ( M ... N ) )
43		elfzle2 10262	. . . . . . . 8 |- ( A e. ( M ... N ) -> A <_ N )
44	42, 43	syl 14	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A <_ N )
45	23, 39, 40, 41, 44	letrd 8302	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ N )
46	38, 45	jca 306	. . . . 5 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( M <_ ( A - 1 ) /\ ( A - 1 ) <_ N ) )
47		elfz2 10249	. . . . 5 |- ( ( A - 1 ) e. ( M ... N ) <-> ( ( M e. ZZ /\ N e. ZZ /\ ( A - 1 ) e. ZZ ) /\ ( M <_ ( A - 1 ) /\ ( A - 1 ) <_ N ) ) )
48	18, 46, 47	sylanbrc 417	. . . 4 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. ( M ... N ) )
49	6, 48	ffvelcdmd 5783	. . 3 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( J ` ( A - 1 ) ) e. ( M ... N ) )
50	1, 20	syl 14	. . . . 5 |- ( ph -> K e. ZZ )
51		zdceq 9554	. . . . 5 |- ( ( A e. ZZ /\ K e. ZZ ) -> DECID A = K )
52	14, 50, 51	syl2anc 411	. . . 4 |- ( ph -> DECID A = K )
53	52	adantr 276	. . 3 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> DECID A = K )
54	2, 49, 53	ifcldadc 3635	. 2 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( M ... N ) )
55	5, 12	ffvelcdmd 5783	. . 3 |- ( ph -> ( J ` A ) e. ( M ... N ) )
56	55	adantr 276	. 2 |- ( ( ph /\ -. A e. ( K ... ( `' J ` K ) ) ) -> ( J ` A ) e. ( M ... N ) )
57		f1ocnv 5596	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
58		f1of 5583	. . . . . 6 |- ( `' J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) --> ( M ... N ) )
59	3, 57, 58	3syl 17	. . . . 5 |- ( ph -> `' J : ( M ... N ) --> ( M ... N ) )
60	59, 1	ffvelcdmd 5783	. . . 4 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
61		elfzelz 10259	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
62	60, 61	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
63		fzdcel 10274	. . 3 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
64	14, 50, 62, 63	syl3anc 1273	. 2 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
65	54, 56, 64	ifcldadc 3635	1 |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   ifcif 3605   class class class wbr 4088   `'ccnv 4724   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   1c1 8032   < clt 8213   <_ cle 8214   - cmin 8349   ZZcz 9478   ...cfz 10242

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243

This theorem is referenced by:  iseqf1olemqval  10761  iseqf1olemqf  10765