Theorem iseqf1olemqcl 10646

Description: Lemma for seq3f1o 10664. (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 5524	. . . . . 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 10148	. . . . . . 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 10147	. . . . . . 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 10149	. . . . . . . . 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 9412	. . . . . . 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 9497	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M e. RR )
20		elfzelz 10149	. . . . . . . . 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 9497	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> K e. RR )
23	17	zred 9497	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
24		elfzle1 10151	. . . . . . . 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 2207	. . . . . . . . . 10 |- ( A = K <-> K = A )
28	26, 27	sylnib 678	. . . . . . . . 9 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> -. K = A )
29		elfzle1 10151	. . . . . . . . . . 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 9421	. . . . . . . . . . 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 9432	. . . . . . . . 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 8198	. . . . . 6 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> M <_ ( A - 1 ) )
39	15	zred 9497	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> A e. RR )
40	11	zred 9497	. . . . . . 7 |- ( ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) /\ -. A = K ) -> N e. RR )
41	39	lem1d 9008	. . . . . . 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 10152	. . . . . . . 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 8198	. . . . . 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 10139	. . . . 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 5718	. . 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 9450	. . . . 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 3600	. 2 |- ( ( ph /\ A e. ( K ... ( `' J ` K ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( M ... N ) )
55	5, 12	ffvelcdmd 5718	. . 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 5537	. . . . . 6 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
58		f1of 5524	. . . . . 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 5718	. . . 4 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
61		elfzelz 10149	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
62	60, 61	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
63		fzdcel 10164	. . 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 3600	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 2176   ifcif 3571   class class class wbr 4045   `'ccnv 4675   -->wf 5268   -1-1-onto->wf1o 5271   ` cfv 5272  (class class class)co 5946   1c1 7928   < clt 8109   <_ cle 8110   - cmin 8245   ZZcz 9374   ...cfz 10132

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133

This theorem is referenced by:  iseqf1olemqval  10647  iseqf1olemqf  10651