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Mirrors > Home > ILE Home > Th. List > fseq1p1m1 | Unicode version |
Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Ref | Expression |
---|---|
fseq1p1m1.1 |
Ref | Expression |
---|---|
fseq1p1m1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 987 | . . . . . 6 | |
2 | nn0p1nn 9016 | . . . . . . . . 9 | |
3 | 2 | adantr 274 | . . . . . . . 8 |
4 | simpr2 988 | . . . . . . . 8 | |
5 | fseq1p1m1.1 | . . . . . . . . 9 | |
6 | fsng 5593 | . . . . . . . . 9 | |
7 | 5, 6 | mpbiri 167 | . . . . . . . 8 |
8 | 3, 4, 7 | syl2anc 408 | . . . . . . 7 |
9 | 4 | snssd 3665 | . . . . . . 7 |
10 | 8, 9 | fssd 5285 | . . . . . 6 |
11 | fzp1disj 9860 | . . . . . . 7 | |
12 | 11 | a1i 9 | . . . . . 6 |
13 | fun2 5296 | . . . . . 6 | |
14 | 1, 10, 12, 13 | syl21anc 1215 | . . . . 5 |
15 | 1z 9080 | . . . . . . . 8 | |
16 | simpl 108 | . . . . . . . . 9 | |
17 | nn0uz 9360 | . . . . . . . . . 10 | |
18 | 1m1e0 8789 | . . . . . . . . . . 11 | |
19 | 18 | fveq2i 5424 | . . . . . . . . . 10 |
20 | 17, 19 | eqtr4i 2163 | . . . . . . . . 9 |
21 | 16, 20 | eleqtrdi 2232 | . . . . . . . 8 |
22 | fzsuc2 9859 | . . . . . . . 8 | |
23 | 15, 21, 22 | sylancr 410 | . . . . . . 7 |
24 | 23 | eqcomd 2145 | . . . . . 6 |
25 | 24 | feq2d 5260 | . . . . 5 |
26 | 14, 25 | mpbid 146 | . . . 4 |
27 | simpr3 989 | . . . . 5 | |
28 | 27 | feq1d 5259 | . . . 4 |
29 | 26, 28 | mpbird 166 | . . 3 |
30 | 27 | reseq1d 4818 | . . . . . 6 |
31 | ffn 5272 | . . . . . . . . . 10 | |
32 | fnresdisj 5233 | . . . . . . . . . 10 | |
33 | 1, 31, 32 | 3syl 17 | . . . . . . . . 9 |
34 | 12, 33 | mpbid 146 | . . . . . . . 8 |
35 | 34 | uneq1d 3229 | . . . . . . 7 |
36 | resundir 4833 | . . . . . . 7 | |
37 | uncom 3220 | . . . . . . . 8 | |
38 | un0 3396 | . . . . . . . 8 | |
39 | 37, 38 | eqtr2i 2161 | . . . . . . 7 |
40 | 35, 36, 39 | 3eqtr4g 2197 | . . . . . 6 |
41 | ffn 5272 | . . . . . . 7 | |
42 | fnresdm 5232 | . . . . . . 7 | |
43 | 10, 41, 42 | 3syl 17 | . . . . . 6 |
44 | 30, 40, 43 | 3eqtrd 2176 | . . . . 5 |
45 | 44 | fveq1d 5423 | . . . 4 |
46 | 16 | nn0zd 9171 | . . . . . 6 |
47 | 46 | peano2zd 9176 | . . . . 5 |
48 | snidg 3554 | . . . . 5 | |
49 | fvres 5445 | . . . . 5 | |
50 | 47, 48, 49 | 3syl 17 | . . . 4 |
51 | 5 | fveq1i 5422 | . . . . . 6 |
52 | fvsng 5616 | . . . . . 6 | |
53 | 51, 52 | syl5eq 2184 | . . . . 5 |
54 | 3, 4, 53 | syl2anc 408 | . . . 4 |
55 | 45, 50, 54 | 3eqtr3d 2180 | . . 3 |
56 | 27 | reseq1d 4818 | . . . 4 |
57 | incom 3268 | . . . . . . . 8 | |
58 | 57, 12 | syl5eq 2184 | . . . . . . 7 |
59 | ffn 5272 | . . . . . . . 8 | |
60 | fnresdisj 5233 | . . . . . . . 8 | |
61 | 8, 59, 60 | 3syl 17 | . . . . . . 7 |
62 | 58, 61 | mpbid 146 | . . . . . 6 |
63 | 62 | uneq2d 3230 | . . . . 5 |
64 | resundir 4833 | . . . . 5 | |
65 | un0 3396 | . . . . . 6 | |
66 | 65 | eqcomi 2143 | . . . . 5 |
67 | 63, 64, 66 | 3eqtr4g 2197 | . . . 4 |
68 | fnresdm 5232 | . . . . 5 | |
69 | 1, 31, 68 | 3syl 17 | . . . 4 |
70 | 56, 67, 69 | 3eqtrrd 2177 | . . 3 |
71 | 29, 55, 70 | 3jca 1161 | . 2 |
72 | simpr1 987 | . . . . 5 | |
73 | fzssp1 9847 | . . . . 5 | |
74 | fssres 5298 | . . . . 5 | |
75 | 72, 73, 74 | sylancl 409 | . . . 4 |
76 | simpr3 989 | . . . . 5 | |
77 | 76 | feq1d 5259 | . . . 4 |
78 | 75, 77 | mpbird 166 | . . 3 |
79 | simpr2 988 | . . . 4 | |
80 | 2 | adantr 274 | . . . . . . 7 |
81 | nnuz 9361 | . . . . . . 7 | |
82 | 80, 81 | eleqtrdi 2232 | . . . . . 6 |
83 | eluzfz2 9812 | . . . . . 6 | |
84 | 82, 83 | syl 14 | . . . . 5 |
85 | 72, 84 | ffvelrnd 5556 | . . . 4 |
86 | 79, 85 | eqeltrrd 2217 | . . 3 |
87 | ffn 5272 | . . . . . . . . 9 | |
88 | 72, 87 | syl 14 | . . . . . . . 8 |
89 | fnressn 5606 | . . . . . . . 8 | |
90 | 88, 84, 89 | syl2anc 408 | . . . . . . 7 |
91 | opeq2 3706 | . . . . . . . . 9 | |
92 | 91 | sneqd 3540 | . . . . . . . 8 |
93 | 79, 92 | syl 14 | . . . . . . 7 |
94 | 90, 93 | eqtrd 2172 | . . . . . 6 |
95 | 94, 5 | syl6reqr 2191 | . . . . 5 |
96 | 76, 95 | uneq12d 3231 | . . . 4 |
97 | simpl 108 | . . . . . . . 8 | |
98 | 97, 20 | eleqtrdi 2232 | . . . . . . 7 |
99 | 15, 98, 22 | sylancr 410 | . . . . . 6 |
100 | 99 | reseq2d 4819 | . . . . 5 |
101 | resundi 4832 | . . . . 5 | |
102 | 100, 101 | syl6req 2189 | . . . 4 |
103 | fnresdm 5232 | . . . . 5 | |
104 | 72, 87, 103 | 3syl 17 | . . . 4 |
105 | 96, 102, 104 | 3eqtrrd 2177 | . . 3 |
106 | 78, 86, 105 | 3jca 1161 | . 2 |
107 | 71, 106 | impbida 585 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cun 3069 cin 3070 wss 3071 c0 3363 csn 3527 cop 3530 cres 4541 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 cc0 7620 c1 7621 caddc 7623 cmin 7933 cn 8720 cn0 8977 cz 9054 cuz 9326 cfz 9790 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 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-apti 7735 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-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-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 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-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 |
This theorem is referenced by: fseq1m1p1 9875 |
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