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 9940 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2220 | . . . . . 6 | |
5 | fveq2 5470 | . . . . . . 7 | |
6 | fveq2 5470 | . . . . . . . 8 | |
7 | 6 | oveq2d 5842 | . . . . . . 7 |
8 | 5, 7 | eqeq12d 2172 | . . . . . 6 |
9 | 4, 8 | imbi12d 233 | . . . . 5 |
10 | 9 | imbi2d 229 | . . . 4 |
11 | eleq1 2220 | . . . . . 6 | |
12 | fveq2 5470 | . . . . . . 7 | |
13 | fveq2 5470 | . . . . . . . 8 | |
14 | 13 | oveq2d 5842 | . . . . . . 7 |
15 | 12, 14 | eqeq12d 2172 | . . . . . 6 |
16 | 11, 15 | imbi12d 233 | . . . . 5 |
17 | 16 | imbi2d 229 | . . . 4 |
18 | eleq1 2220 | . . . . . 6 | |
19 | fveq2 5470 | . . . . . . 7 | |
20 | fveq2 5470 | . . . . . . . 8 | |
21 | 20 | oveq2d 5842 | . . . . . . 7 |
22 | 19, 21 | eqeq12d 2172 | . . . . . 6 |
23 | 18, 22 | imbi12d 233 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | eleq1 2220 | . . . . . 6 | |
26 | fveq2 5470 | . . . . . . 7 | |
27 | fveq2 5470 | . . . . . . . 8 | |
28 | 27 | oveq2d 5842 | . . . . . . 7 |
29 | 26, 28 | eqeq12d 2172 | . . . . . 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 10370 | . . . . . 6 |
36 | eluzel2 9449 | . . . . . . . . 9 | |
37 | 1, 36 | syl 14 | . . . . . . . 8 |
38 | simpl 108 | . . . . . . . . 9 | |
39 | eluzel2 9449 | . . . . . . . . . . . 12 | |
40 | 32, 39 | syl 14 | . . . . . . . . . . 11 |
41 | 40 | adantr 274 | . . . . . . . . . 10 |
42 | eluzelz 9453 | . . . . . . . . . . 11 | |
43 | 42 | adantl 275 | . . . . . . . . . 10 |
44 | 41 | zred 9291 | . . . . . . . . . . 11 |
45 | eluzelz 9453 | . . . . . . . . . . . . . 14 | |
46 | 32, 45 | syl 14 | . . . . . . . . . . . . 13 |
47 | 46 | zred 9291 | . . . . . . . . . . . 12 |
48 | 47 | adantr 274 | . . . . . . . . . . 11 |
49 | 43 | zred 9291 | . . . . . . . . . . 11 |
50 | eluzle 9456 | . . . . . . . . . . . . 13 | |
51 | 32, 50 | syl 14 | . . . . . . . . . . . 12 |
52 | 51 | adantr 274 | . . . . . . . . . . 11 |
53 | peano2re 8015 | . . . . . . . . . . . . 13 | |
54 | 48, 53 | syl 14 | . . . . . . . . . . . 12 |
55 | 48 | lep1d 8807 | . . . . . . . . . . . 12 |
56 | eluzle 9456 | . . . . . . . . . . . . 13 | |
57 | 56 | adantl 275 | . . . . . . . . . . . 12 |
58 | 48, 54, 49, 55, 57 | letrd 8003 | . . . . . . . . . . 11 |
59 | 44, 48, 49, 52, 58 | letrd 8003 | . . . . . . . . . 10 |
60 | eluz2 9450 | . . . . . . . . . 10 | |
61 | 41, 43, 59, 60 | syl3anbrc 1166 | . . . . . . . . 9 |
62 | 38, 61, 33 | syl2anc 409 | . . . . . . . 8 |
63 | 37, 62, 34 | seq3-1 10368 | . . . . . . 7 |
64 | 63 | oveq2d 5842 | . . . . . 6 |
65 | 35, 64 | eqtr4d 2193 | . . . . 5 |
66 | 65 | a1i13 24 | . . . 4 |
67 | peano2fzr 9945 | . . . . . . . 8 | |
68 | 67 | adantl 275 | . . . . . . 7 |
69 | 68 | expr 373 | . . . . . 6 |
70 | 69 | imim1d 75 | . . . . 5 |
71 | oveq1 5833 | . . . . . 6 | |
72 | simprl 521 | . . . . . . . . 9 | |
73 | peano2uz 9499 | . . . . . . . . . . 11 | |
74 | 32, 73 | syl 14 | . . . . . . . . . 10 |
75 | 74 | adantr 274 | . . . . . . . . 9 |
76 | uztrn 9460 | . . . . . . . . 9 | |
77 | 72, 75, 76 | syl2anc 409 | . . . . . . . 8 |
78 | 33 | adantlr 469 | . . . . . . . 8 |
79 | 34 | adantlr 469 | . . . . . . . 8 |
80 | 77, 78, 79 | seq3p1 10370 | . . . . . . 7 |
81 | 62 | adantlr 469 | . . . . . . . . . 10 |
82 | 72, 81, 79 | seq3p1 10370 | . . . . . . . . 9 |
83 | 82 | oveq2d 5842 | . . . . . . . 8 |
84 | simpl 108 | . . . . . . . . 9 | |
85 | eqid 2157 | . . . . . . . . . . . 12 | |
86 | 85, 40, 33, 34 | seqf 10369 | . . . . . . . . . . 11 |
87 | 86, 32 | ffvelrnd 5605 | . . . . . . . . . 10 |
88 | 87 | adantr 274 | . . . . . . . . 9 |
89 | eqid 2157 | . . . . . . . . . . 11 | |
90 | 37 | adantr 274 | . . . . . . . . . . 11 |
91 | 89, 90, 81, 79 | seqf 10369 | . . . . . . . . . 10 |
92 | 91, 72 | ffvelrnd 5605 | . . . . . . . . 9 |
93 | fveq2 5470 | . . . . . . . . . . 11 | |
94 | 93 | eleq1d 2226 | . . . . . . . . . 10 |
95 | 33 | ralrimiva 2530 | . . . . . . . . . . 11 |
96 | 95 | adantr 274 | . . . . . . . . . 10 |
97 | fzssuz 9973 | . . . . . . . . . . . 12 | |
98 | uzss 9464 | . . . . . . . . . . . . 13 | |
99 | 74, 98 | syl 14 | . . . . . . . . . . . 12 |
100 | 97, 99 | sstrid 3139 | . . . . . . . . . . 11 |
101 | simpr 109 | . . . . . . . . . . 11 | |
102 | ssel2 3123 | . . . . . . . . . . 11 | |
103 | 100, 101, 102 | syl2an 287 | . . . . . . . . . 10 |
104 | 94, 96, 103 | rspcdva 2821 | . . . . . . . . 9 |
105 | seq3split.2 | . . . . . . . . . 10 | |
106 | 105 | caovassg 5981 | . . . . . . . . 9 |
107 | 84, 88, 92, 104, 106 | syl13anc 1222 | . . . . . . . 8 |
108 | 83, 107 | eqtr4d 2193 | . . . . . . 7 |
109 | 80, 108 | eqeq12d 2172 | . . . . . 6 |
110 | 71, 109 | syl5ibr 155 | . . . . 5 |
111 | 70, 110 | animpimp2impd 549 | . . . 4 |
112 | 10, 17, 24, 31, 66, 111 | uzind4 9504 | . . 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 963 wceq 1335 wcel 2128 wral 2435 wss 3102 class class class wbr 3967 cfv 5172 (class class class)co 5826 cr 7733 c1 7735 caddc 7737 cle 7915 cz 9172 cuz 9444 cfz 9918 cseq 10353 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-frec 6340 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-n0 9096 df-z 9173 df-uz 9445 df-fz 9919 df-seqfrec 10354 |
This theorem is referenced by: seq3-1p 10388 seq3f1olemqsumk 10407 seq3f1olemqsum 10408 bcval5 10648 clim2ser 11245 clim2ser2 11246 isumsplit 11399 cvgratnnlemseq 11434 clim2divap 11448 |
Copyright terms: Public domain | W3C validator |