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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 9009 | . . . . . . . . 9 | |
3 | 2 | adantr 274 | . . . . . . . 8 |
4 | simpr2 988 | . . . . . . . 8 | |
5 | fseq1p1m1.1 | . . . . . . . . 9 | |
6 | fsng 5586 | . . . . . . . . 9 | |
7 | 5, 6 | mpbiri 167 | . . . . . . . 8 |
8 | 3, 4, 7 | syl2anc 408 | . . . . . . 7 |
9 | 4 | snssd 3660 | . . . . . . 7 |
10 | 8, 9 | fssd 5280 | . . . . . 6 |
11 | fzp1disj 9853 | . . . . . . 7 | |
12 | 11 | a1i 9 | . . . . . 6 |
13 | fun2 5291 | . . . . . 6 | |
14 | 1, 10, 12, 13 | syl21anc 1215 | . . . . 5 |
15 | 1z 9073 | . . . . . . . 8 | |
16 | simpl 108 | . . . . . . . . 9 | |
17 | nn0uz 9353 | . . . . . . . . . 10 | |
18 | 1m1e0 8782 | . . . . . . . . . . 11 | |
19 | 18 | fveq2i 5417 | . . . . . . . . . 10 |
20 | 17, 19 | eqtr4i 2161 | . . . . . . . . 9 |
21 | 16, 20 | eleqtrdi 2230 | . . . . . . . 8 |
22 | fzsuc2 9852 | . . . . . . . 8 | |
23 | 15, 21, 22 | sylancr 410 | . . . . . . 7 |
24 | 23 | eqcomd 2143 | . . . . . 6 |
25 | 24 | feq2d 5255 | . . . . 5 |
26 | 14, 25 | mpbid 146 | . . . 4 |
27 | simpr3 989 | . . . . 5 | |
28 | 27 | feq1d 5254 | . . . 4 |
29 | 26, 28 | mpbird 166 | . . 3 |
30 | 27 | reseq1d 4813 | . . . . . 6 |
31 | ffn 5267 | . . . . . . . . . 10 | |
32 | fnresdisj 5228 | . . . . . . . . . 10 | |
33 | 1, 31, 32 | 3syl 17 | . . . . . . . . 9 |
34 | 12, 33 | mpbid 146 | . . . . . . . 8 |
35 | 34 | uneq1d 3224 | . . . . . . 7 |
36 | resundir 4828 | . . . . . . 7 | |
37 | uncom 3215 | . . . . . . . 8 | |
38 | un0 3391 | . . . . . . . 8 | |
39 | 37, 38 | eqtr2i 2159 | . . . . . . 7 |
40 | 35, 36, 39 | 3eqtr4g 2195 | . . . . . 6 |
41 | ffn 5267 | . . . . . . 7 | |
42 | fnresdm 5227 | . . . . . . 7 | |
43 | 10, 41, 42 | 3syl 17 | . . . . . 6 |
44 | 30, 40, 43 | 3eqtrd 2174 | . . . . 5 |
45 | 44 | fveq1d 5416 | . . . 4 |
46 | 16 | nn0zd 9164 | . . . . . 6 |
47 | 46 | peano2zd 9169 | . . . . 5 |
48 | snidg 3549 | . . . . 5 | |
49 | fvres 5438 | . . . . 5 | |
50 | 47, 48, 49 | 3syl 17 | . . . 4 |
51 | 5 | fveq1i 5415 | . . . . . 6 |
52 | fvsng 5609 | . . . . . 6 | |
53 | 51, 52 | syl5eq 2182 | . . . . 5 |
54 | 3, 4, 53 | syl2anc 408 | . . . 4 |
55 | 45, 50, 54 | 3eqtr3d 2178 | . . 3 |
56 | 27 | reseq1d 4813 | . . . 4 |
57 | incom 3263 | . . . . . . . 8 | |
58 | 57, 12 | syl5eq 2182 | . . . . . . 7 |
59 | ffn 5267 | . . . . . . . 8 | |
60 | fnresdisj 5228 | . . . . . . . 8 | |
61 | 8, 59, 60 | 3syl 17 | . . . . . . 7 |
62 | 58, 61 | mpbid 146 | . . . . . 6 |
63 | 62 | uneq2d 3225 | . . . . 5 |
64 | resundir 4828 | . . . . 5 | |
65 | un0 3391 | . . . . . 6 | |
66 | 65 | eqcomi 2141 | . . . . 5 |
67 | 63, 64, 66 | 3eqtr4g 2195 | . . . 4 |
68 | fnresdm 5227 | . . . . 5 | |
69 | 1, 31, 68 | 3syl 17 | . . . 4 |
70 | 56, 67, 69 | 3eqtrrd 2175 | . . 3 |
71 | 29, 55, 70 | 3jca 1161 | . 2 |
72 | simpr1 987 | . . . . 5 | |
73 | fzssp1 9840 | . . . . 5 | |
74 | fssres 5293 | . . . . 5 | |
75 | 72, 73, 74 | sylancl 409 | . . . 4 |
76 | simpr3 989 | . . . . 5 | |
77 | 76 | feq1d 5254 | . . . 4 |
78 | 75, 77 | mpbird 166 | . . 3 |
79 | simpr2 988 | . . . 4 | |
80 | 2 | adantr 274 | . . . . . . 7 |
81 | nnuz 9354 | . . . . . . 7 | |
82 | 80, 81 | eleqtrdi 2230 | . . . . . 6 |
83 | eluzfz2 9805 | . . . . . 6 | |
84 | 82, 83 | syl 14 | . . . . 5 |
85 | 72, 84 | ffvelrnd 5549 | . . . 4 |
86 | 79, 85 | eqeltrrd 2215 | . . 3 |
87 | ffn 5267 | . . . . . . . . 9 | |
88 | 72, 87 | syl 14 | . . . . . . . 8 |
89 | fnressn 5599 | . . . . . . . 8 | |
90 | 88, 84, 89 | syl2anc 408 | . . . . . . 7 |
91 | opeq2 3701 | . . . . . . . . 9 | |
92 | 91 | sneqd 3535 | . . . . . . . 8 |
93 | 79, 92 | syl 14 | . . . . . . 7 |
94 | 90, 93 | eqtrd 2170 | . . . . . 6 |
95 | 94, 5 | syl6reqr 2189 | . . . . 5 |
96 | 76, 95 | uneq12d 3226 | . . . 4 |
97 | simpl 108 | . . . . . . . 8 | |
98 | 97, 20 | eleqtrdi 2230 | . . . . . . 7 |
99 | 15, 98, 22 | sylancr 410 | . . . . . 6 |
100 | 99 | reseq2d 4814 | . . . . 5 |
101 | resundi 4827 | . . . . 5 | |
102 | 100, 101 | syl6req 2187 | . . . 4 |
103 | fnresdm 5227 | . . . . 5 | |
104 | 72, 87, 103 | 3syl 17 | . . . 4 |
105 | 96, 102, 104 | 3eqtrrd 2175 | . . 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 3064 cin 3065 wss 3066 c0 3358 csn 3522 cop 3525 cres 4536 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cc0 7613 c1 7614 caddc 7616 cmin 7926 cn 8713 cn0 8970 cz 9047 cuz 9319 cfz 9783 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: fseq1m1p1 9868 |
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