Theorem iseqf1olemqcl 10824

Description: Lemma for seq3f1o 10842. (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 5592	. . . . . 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 10321	. . . . . . 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 10320	. . . . . . 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 10322	. . . . . . . . 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 9578	. . . . . . 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 1204	. . . . 5 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( M e. ZZ /\ N e. ZZ /\ ( A - 1 ) e. ZZ ) )
19	9	zred 9663	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. RR )
20		elfzelz 10322	. . . . . . . . 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 9663	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. RR )
23	17	zred 9663	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
24		elfzle1 10324	. . . . . . . 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 683	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. K = A )
29		elfzle1 10324	. . . . . . . . . . 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 9587	. . . . . . . . . . 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 1386	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K < A )
35		zltlem1 9598	. . . . . . . . 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 8362	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ ( A - 1 ) )
39	15	zred 9663	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. RR )
40	11	zred 9663	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. RR )
41	39	lem1d 9172	. . . . . . 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 10325	. . . . . . . 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 8362	. . . . . 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 10312	. . . . 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 5791	. . 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 9616	. . . . 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 3639	. 2 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( M ... N ) )
55	5, 12	ffvelcdmd 5791	. . 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 5605	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
58		f1of 5592	. . . . . 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 5791	. . . 4 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
61		elfzelz 10322	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
62	60, 61	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
63		fzdcel 10337	. . 3 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
64	14, 50, 62, 63	syl3anc 1274	. 2 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
65	54, 56, 64	ifcldadc 3639	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   ifcif 3607   class class class wbr 4093   `'ccnv 4730   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   1c1 8093   < clt 8273   <_ cle 8274   - cmin 8409   ZZcz 9540   ...cfz 10305

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by:  iseqf1olemqval  10825  iseqf1olemqf  10829