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Mirrors > Home > ILE Home > Th. List > seq3split | Unicode version |
Description: Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.) |
Ref | Expression |
---|---|
seq3split.1 | |
seq3split.2 | |
seq3split.3 | |
seq3split.4 | |
seq3split.5 |
Ref | Expression |
---|---|
seq3split |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3split.3 | . . 3 | |
2 | eluzfz2 9812 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2202 | . . . . . 6 | |
5 | fveq2 5421 | . . . . . . 7 | |
6 | fveq2 5421 | . . . . . . . 8 | |
7 | 6 | oveq2d 5790 | . . . . . . 7 |
8 | 5, 7 | eqeq12d 2154 | . . . . . 6 |
9 | 4, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | imbi2d 229 | . . . 4 |
11 | eleq1 2202 | . . . . . 6 | |
12 | fveq2 5421 | . . . . . . 7 | |
13 | fveq2 5421 | . . . . . . . 8 | |
14 | 13 | oveq2d 5790 | . . . . . . 7 |
15 | 12, 14 | eqeq12d 2154 | . . . . . 6 |
16 | 11, 15 | imbi12d 233 | . . . . 5 |
17 | 16 | imbi2d 229 | . . . 4 |
18 | eleq1 2202 | . . . . . 6 | |
19 | fveq2 5421 | . . . . . . 7 | |
20 | fveq2 5421 | . . . . . . . 8 | |
21 | 20 | oveq2d 5790 | . . . . . . 7 |
22 | 19, 21 | eqeq12d 2154 | . . . . . 6 |
23 | 18, 22 | imbi12d 233 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | eleq1 2202 | . . . . . 6 | |
26 | fveq2 5421 | . . . . . . 7 | |
27 | fveq2 5421 | . . . . . . . 8 | |
28 | 27 | oveq2d 5790 | . . . . . . 7 |
29 | 26, 28 | eqeq12d 2154 | . . . . . 6 |
30 | 25, 29 | imbi12d 233 | . . . . 5 |
31 | 30 | imbi2d 229 | . . . 4 |
32 | seq3split.4 | . . . . . . 7 | |
33 | seq3split.5 | . . . . . . 7 | |
34 | seq3split.1 | . . . . . . 7 | |
35 | 32, 33, 34 | seq3p1 10235 | . . . . . 6 |
36 | eluzel2 9331 | . . . . . . . . 9 | |
37 | 1, 36 | syl 14 | . . . . . . . 8 |
38 | simpl 108 | . . . . . . . . 9 | |
39 | eluzel2 9331 | . . . . . . . . . . . 12 | |
40 | 32, 39 | syl 14 | . . . . . . . . . . 11 |
41 | 40 | adantr 274 | . . . . . . . . . 10 |
42 | eluzelz 9335 | . . . . . . . . . . 11 | |
43 | 42 | adantl 275 | . . . . . . . . . 10 |
44 | 41 | zred 9173 | . . . . . . . . . . 11 |
45 | eluzelz 9335 | . . . . . . . . . . . . . 14 | |
46 | 32, 45 | syl 14 | . . . . . . . . . . . . 13 |
47 | 46 | zred 9173 | . . . . . . . . . . . 12 |
48 | 47 | adantr 274 | . . . . . . . . . . 11 |
49 | 43 | zred 9173 | . . . . . . . . . . 11 |
50 | eluzle 9338 | . . . . . . . . . . . . 13 | |
51 | 32, 50 | syl 14 | . . . . . . . . . . . 12 |
52 | 51 | adantr 274 | . . . . . . . . . . 11 |
53 | peano2re 7898 | . . . . . . . . . . . . 13 | |
54 | 48, 53 | syl 14 | . . . . . . . . . . . 12 |
55 | 48 | lep1d 8689 | . . . . . . . . . . . 12 |
56 | eluzle 9338 | . . . . . . . . . . . . 13 | |
57 | 56 | adantl 275 | . . . . . . . . . . . 12 |
58 | 48, 54, 49, 55, 57 | letrd 7886 | . . . . . . . . . . 11 |
59 | 44, 48, 49, 52, 58 | letrd 7886 | . . . . . . . . . 10 |
60 | eluz2 9332 | . . . . . . . . . 10 | |
61 | 41, 43, 59, 60 | syl3anbrc 1165 | . . . . . . . . 9 |
62 | 38, 61, 33 | syl2anc 408 | . . . . . . . 8 |
63 | 37, 62, 34 | seq3-1 10233 | . . . . . . 7 |
64 | 63 | oveq2d 5790 | . . . . . 6 |
65 | 35, 64 | eqtr4d 2175 | . . . . 5 |
66 | 65 | a1i13 24 | . . . 4 |
67 | peano2fzr 9817 | . . . . . . . 8 | |
68 | 67 | adantl 275 | . . . . . . 7 |
69 | 68 | expr 372 | . . . . . 6 |
70 | 69 | imim1d 75 | . . . . 5 |
71 | oveq1 5781 | . . . . . 6 | |
72 | simprl 520 | . . . . . . . . 9 | |
73 | peano2uz 9378 | . . . . . . . . . . 11 | |
74 | 32, 73 | syl 14 | . . . . . . . . . 10 |
75 | 74 | adantr 274 | . . . . . . . . 9 |
76 | uztrn 9342 | . . . . . . . . 9 | |
77 | 72, 75, 76 | syl2anc 408 | . . . . . . . 8 |
78 | 33 | adantlr 468 | . . . . . . . 8 |
79 | 34 | adantlr 468 | . . . . . . . 8 |
80 | 77, 78, 79 | seq3p1 10235 | . . . . . . 7 |
81 | 62 | adantlr 468 | . . . . . . . . . 10 |
82 | 72, 81, 79 | seq3p1 10235 | . . . . . . . . 9 |
83 | 82 | oveq2d 5790 | . . . . . . . 8 |
84 | simpl 108 | . . . . . . . . 9 | |
85 | eqid 2139 | . . . . . . . . . . . 12 | |
86 | 85, 40, 33, 34 | seqf 10234 | . . . . . . . . . . 11 |
87 | 86, 32 | ffvelrnd 5556 | . . . . . . . . . 10 |
88 | 87 | adantr 274 | . . . . . . . . 9 |
89 | eqid 2139 | . . . . . . . . . . 11 | |
90 | 37 | adantr 274 | . . . . . . . . . . 11 |
91 | 89, 90, 81, 79 | seqf 10234 | . . . . . . . . . 10 |
92 | 91, 72 | ffvelrnd 5556 | . . . . . . . . 9 |
93 | fveq2 5421 | . . . . . . . . . . 11 | |
94 | 93 | eleq1d 2208 | . . . . . . . . . 10 |
95 | 33 | ralrimiva 2505 | . . . . . . . . . . 11 |
96 | 95 | adantr 274 | . . . . . . . . . 10 |
97 | fzssuz 9845 | . . . . . . . . . . . 12 | |
98 | uzss 9346 | . . . . . . . . . . . . 13 | |
99 | 74, 98 | syl 14 | . . . . . . . . . . . 12 |
100 | 97, 99 | sstrid 3108 | . . . . . . . . . . 11 |
101 | simpr 109 | . . . . . . . . . . 11 | |
102 | ssel2 3092 | . . . . . . . . . . 11 | |
103 | 100, 101, 102 | syl2an 287 | . . . . . . . . . 10 |
104 | 94, 96, 103 | rspcdva 2794 | . . . . . . . . 9 |
105 | seq3split.2 | . . . . . . . . . 10 | |
106 | 105 | caovassg 5929 | . . . . . . . . 9 |
107 | 84, 88, 92, 104, 106 | syl13anc 1218 | . . . . . . . 8 |
108 | 83, 107 | eqtr4d 2175 | . . . . . . 7 |
109 | 80, 108 | eqeq12d 2154 | . . . . . 6 |
110 | 71, 109 | syl5ibr 155 | . . . . 5 |
111 | 70, 110 | animpimp2impd 548 | . . . 4 |
112 | 10, 17, 24, 31, 66, 111 | uzind4 9383 | . . 3 |
113 | 1, 112 | mpcom 36 | . 2 |
114 | 3, 113 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 wss 3071 class class class wbr 3929 cfv 5123 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 cle 7801 cz 9054 cuz 9326 cfz 9790 cseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-seqfrec 10219 |
This theorem is referenced by: seq3-1p 10253 seq3f1olemqsumk 10272 seq3f1olemqsum 10273 bcval5 10509 clim2ser 11106 clim2ser2 11107 isumsplit 11260 cvgratnnlemseq 11295 clim2divap 11309 |
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