Theorem iseqf1olemqcl 10486

Description: Lemma for seq3f1o 10504. (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 5462	. . . . . 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 10024	. . . . . . 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 10023	. . . . . . 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 10025	. . . . . . . . 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 9291	. . . . . . 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 1177	. . . . 5 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( M e. ZZ /\ N e. ZZ /\ ( A - 1 ) e. ZZ ) )
19	9	zred 9375	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. RR )
20		elfzelz 10025	. . . . . . . . 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 9375	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. RR )
23	17	zred 9375	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
24		elfzle1 10027	. . . . . . . 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 2179	. . . . . . . . . 10 |- ( A = K <-> K = A )
28	26, 27	sylnib 676	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. K = A )
29		elfzle1 10027	. . . . . . . . . . 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 9300	. . . . . . . . . . 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 1349	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K < A )
35		zltlem1 9310	. . . . . . . . 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 8081	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ ( A - 1 ) )
39	15	zred 9375	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. RR )
40	11	zred 9375	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. RR )
41	39	lem1d 8890	. . . . . . 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 10028	. . . . . . . 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 8081	. . . . . 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 10015	. . . . 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 5653	. . 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 9328	. . . . 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 3564	. 2 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( M ... N ) )
55	5, 12	ffvelcdmd 5653	. . 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 5475	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
58		f1of 5462	. . . . . 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 5653	. . . 4 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
61		elfzelz 10025	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
62	60, 61	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
63		fzdcel 10040	. . 3 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
64	14, 50, 62, 63	syl3anc 1238	. 2 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
65	54, 56, 64	ifcldadc 3564	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 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   ifcif 3535   class class class wbr 4004   `'ccnv 4626   -->wf 5213   -1-1-onto->wf1o 5216   ` cfv 5217  (class class class)co 5875   1c1 7812   < clt 7992   <_ cle 7993   - cmin 8128   ZZcz 9253   ...cfz 10008

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009

This theorem is referenced by:  iseqf1olemqval  10487  iseqf1olemqf  10491