Theorem iseqf1olemqcl 10681

Description: Lemma for seq3f1o 10699. (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 5544	. . . . . 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 10181	. . . . . . 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 10180	. . . . . . 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 10182	. . . . . . . . 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 9445	. . . . . . 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 1180	. . . . 5 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( M e. ZZ /\ N e. ZZ /\ ( A - 1 ) e. ZZ ) )
19	9	zred 9530	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. RR )
20		elfzelz 10182	. . . . . . . . 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 9530	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. RR )
23	17	zred 9530	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
24		elfzle1 10184	. . . . . . . 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 2209	. . . . . . . . . 10 |- ( A = K <-> K = A )
28	26, 27	sylnib 678	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. K = A )
29		elfzle1 10184	. . . . . . . . . . 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 9454	. . . . . . . . . . 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 1362	. . . . . . . 8 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K < A )
35		zltlem1 9465	. . . . . . . . 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 8231	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ ( A - 1 ) )
39	15	zred 9530	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. RR )
40	11	zred 9530	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. RR )
41	39	lem1d 9041	. . . . . . 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 10185	. . . . . . . 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 8231	. . . . . 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 10172	. . . . 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 5739	. . 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 9483	. . . . 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 3609	. 2 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( M ... N ) )
55	5, 12	ffvelcdmd 5739	. . 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 5557	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
58		f1of 5544	. . . . . 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 5739	. . . 4 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
61		elfzelz 10182	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
62	60, 61	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
63		fzdcel 10197	. . 3 |- ( ( A e. ZZ /\ K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> DECID A e. ( K ... ( `' J ` K ) ) )
64	14, 50, 62, 63	syl3anc 1250	. 2 |- ( ph -> DECID A e. ( K ... ( `' J ` K ) ) )
65	54, 56, 64	ifcldadc 3609	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2178   ifcif 3579   class class class wbr 4059   `'ccnv 4692   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   1c1 7961   < clt 8142   <_ cle 8143   - cmin 8278   ZZcz 9407   ...cfz 10165

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-dc 837  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-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  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-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

This theorem is referenced by:  iseqf1olemqval  10682  iseqf1olemqf  10686