Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9988 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2233 | . . . . . 6 | |
5 | fveq2 5496 | . . . . . . 7 | |
6 | fveq2 5496 | . . . . . . . 8 | |
7 | 6 | oveq2d 5869 | . . . . . . 7 |
8 | 5, 7 | eqeq12d 2185 | . . . . . 6 |
9 | 4, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | imbi2d 229 | . . . 4 |
11 | eleq1 2233 | . . . . . 6 | |
12 | fveq2 5496 | . . . . . . 7 | |
13 | fveq2 5496 | . . . . . . . 8 | |
14 | 13 | oveq2d 5869 | . . . . . . 7 |
15 | 12, 14 | eqeq12d 2185 | . . . . . 6 |
16 | 11, 15 | imbi12d 233 | . . . . 5 |
17 | 16 | imbi2d 229 | . . . 4 |
18 | eleq1 2233 | . . . . . 6 | |
19 | fveq2 5496 | . . . . . . 7 | |
20 | fveq2 5496 | . . . . . . . 8 | |
21 | 20 | oveq2d 5869 | . . . . . . 7 |
22 | 19, 21 | eqeq12d 2185 | . . . . . 6 |
23 | 18, 22 | imbi12d 233 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | eleq1 2233 | . . . . . 6 | |
26 | fveq2 5496 | . . . . . . 7 | |
27 | fveq2 5496 | . . . . . . . 8 | |
28 | 27 | oveq2d 5869 | . . . . . . 7 |
29 | 26, 28 | eqeq12d 2185 | . . . . . 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 10418 | . . . . . 6 |
36 | eluzel2 9492 | . . . . . . . . 9 | |
37 | 1, 36 | syl 14 | . . . . . . . 8 |
38 | simpl 108 | . . . . . . . . 9 | |
39 | eluzel2 9492 | . . . . . . . . . . . 12 | |
40 | 32, 39 | syl 14 | . . . . . . . . . . 11 |
41 | 40 | adantr 274 | . . . . . . . . . 10 |
42 | eluzelz 9496 | . . . . . . . . . . 11 | |
43 | 42 | adantl 275 | . . . . . . . . . 10 |
44 | 41 | zred 9334 | . . . . . . . . . . 11 |
45 | eluzelz 9496 | . . . . . . . . . . . . . 14 | |
46 | 32, 45 | syl 14 | . . . . . . . . . . . . 13 |
47 | 46 | zred 9334 | . . . . . . . . . . . 12 |
48 | 47 | adantr 274 | . . . . . . . . . . 11 |
49 | 43 | zred 9334 | . . . . . . . . . . 11 |
50 | eluzle 9499 | . . . . . . . . . . . . 13 | |
51 | 32, 50 | syl 14 | . . . . . . . . . . . 12 |
52 | 51 | adantr 274 | . . . . . . . . . . 11 |
53 | peano2re 8055 | . . . . . . . . . . . . 13 | |
54 | 48, 53 | syl 14 | . . . . . . . . . . . 12 |
55 | 48 | lep1d 8847 | . . . . . . . . . . . 12 |
56 | eluzle 9499 | . . . . . . . . . . . . 13 | |
57 | 56 | adantl 275 | . . . . . . . . . . . 12 |
58 | 48, 54, 49, 55, 57 | letrd 8043 | . . . . . . . . . . 11 |
59 | 44, 48, 49, 52, 58 | letrd 8043 | . . . . . . . . . 10 |
60 | eluz2 9493 | . . . . . . . . . 10 | |
61 | 41, 43, 59, 60 | syl3anbrc 1176 | . . . . . . . . 9 |
62 | 38, 61, 33 | syl2anc 409 | . . . . . . . 8 |
63 | 37, 62, 34 | seq3-1 10416 | . . . . . . 7 |
64 | 63 | oveq2d 5869 | . . . . . 6 |
65 | 35, 64 | eqtr4d 2206 | . . . . 5 |
66 | 65 | a1i13 24 | . . . 4 |
67 | peano2fzr 9993 | . . . . . . . 8 | |
68 | 67 | adantl 275 | . . . . . . 7 |
69 | 68 | expr 373 | . . . . . 6 |
70 | 69 | imim1d 75 | . . . . 5 |
71 | oveq1 5860 | . . . . . 6 | |
72 | simprl 526 | . . . . . . . . 9 | |
73 | peano2uz 9542 | . . . . . . . . . . 11 | |
74 | 32, 73 | syl 14 | . . . . . . . . . 10 |
75 | 74 | adantr 274 | . . . . . . . . 9 |
76 | uztrn 9503 | . . . . . . . . 9 | |
77 | 72, 75, 76 | syl2anc 409 | . . . . . . . 8 |
78 | 33 | adantlr 474 | . . . . . . . 8 |
79 | 34 | adantlr 474 | . . . . . . . 8 |
80 | 77, 78, 79 | seq3p1 10418 | . . . . . . 7 |
81 | 62 | adantlr 474 | . . . . . . . . . 10 |
82 | 72, 81, 79 | seq3p1 10418 | . . . . . . . . 9 |
83 | 82 | oveq2d 5869 | . . . . . . . 8 |
84 | simpl 108 | . . . . . . . . 9 | |
85 | eqid 2170 | . . . . . . . . . . . 12 | |
86 | 85, 40, 33, 34 | seqf 10417 | . . . . . . . . . . 11 |
87 | 86, 32 | ffvelrnd 5632 | . . . . . . . . . 10 |
88 | 87 | adantr 274 | . . . . . . . . 9 |
89 | eqid 2170 | . . . . . . . . . . 11 | |
90 | 37 | adantr 274 | . . . . . . . . . . 11 |
91 | 89, 90, 81, 79 | seqf 10417 | . . . . . . . . . 10 |
92 | 91, 72 | ffvelrnd 5632 | . . . . . . . . 9 |
93 | fveq2 5496 | . . . . . . . . . . 11 | |
94 | 93 | eleq1d 2239 | . . . . . . . . . 10 |
95 | 33 | ralrimiva 2543 | . . . . . . . . . . 11 |
96 | 95 | adantr 274 | . . . . . . . . . 10 |
97 | fzssuz 10021 | . . . . . . . . . . . 12 | |
98 | uzss 9507 | . . . . . . . . . . . . 13 | |
99 | 74, 98 | syl 14 | . . . . . . . . . . . 12 |
100 | 97, 99 | sstrid 3158 | . . . . . . . . . . 11 |
101 | simpr 109 | . . . . . . . . . . 11 | |
102 | ssel2 3142 | . . . . . . . . . . 11 | |
103 | 100, 101, 102 | syl2an 287 | . . . . . . . . . 10 |
104 | 94, 96, 103 | rspcdva 2839 | . . . . . . . . 9 |
105 | seq3split.2 | . . . . . . . . . 10 | |
106 | 105 | caovassg 6011 | . . . . . . . . 9 |
107 | 84, 88, 92, 104, 106 | syl13anc 1235 | . . . . . . . 8 |
108 | 83, 107 | eqtr4d 2206 | . . . . . . 7 |
109 | 80, 108 | eqeq12d 2185 | . . . . . 6 |
110 | 71, 109 | syl5ibr 155 | . . . . 5 |
111 | 70, 110 | animpimp2impd 554 | . . . 4 |
112 | 10, 17, 24, 31, 66, 111 | uzind4 9547 | . . 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 973 wceq 1348 wcel 2141 wral 2448 wss 3121 class class class wbr 3989 cfv 5198 (class class class)co 5853 cr 7773 c1 7775 caddc 7777 cle 7955 cz 9212 cuz 9487 cfz 9965 cseq 10401 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-seqfrec 10402 |
This theorem is referenced by: seq3-1p 10436 seq3f1olemqsumk 10455 seq3f1olemqsum 10456 bcval5 10697 clim2ser 11300 clim2ser2 11301 isumsplit 11454 cvgratnnlemseq 11489 clim2divap 11503 |
Copyright terms: Public domain | W3C validator |