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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 992 | . . . . . 6 | |
2 | nn0p1nn 9144 | . . . . . . . . 9 | |
3 | 2 | adantr 274 | . . . . . . . 8 |
4 | simpr2 993 | . . . . . . . 8 | |
5 | fseq1p1m1.1 | . . . . . . . . 9 | |
6 | fsng 5652 | . . . . . . . . 9 | |
7 | 5, 6 | mpbiri 167 | . . . . . . . 8 |
8 | 3, 4, 7 | syl2anc 409 | . . . . . . 7 |
9 | 4 | snssd 3712 | . . . . . . 7 |
10 | 8, 9 | fssd 5344 | . . . . . 6 |
11 | fzp1disj 10005 | . . . . . . 7 | |
12 | 11 | a1i 9 | . . . . . 6 |
13 | fun2 5355 | . . . . . 6 | |
14 | 1, 10, 12, 13 | syl21anc 1226 | . . . . 5 |
15 | 1z 9208 | . . . . . . . 8 | |
16 | simpl 108 | . . . . . . . . 9 | |
17 | nn0uz 9491 | . . . . . . . . . 10 | |
18 | 1m1e0 8917 | . . . . . . . . . . 11 | |
19 | 18 | fveq2i 5483 | . . . . . . . . . 10 |
20 | 17, 19 | eqtr4i 2188 | . . . . . . . . 9 |
21 | 16, 20 | eleqtrdi 2257 | . . . . . . . 8 |
22 | fzsuc2 10004 | . . . . . . . 8 | |
23 | 15, 21, 22 | sylancr 411 | . . . . . . 7 |
24 | 23 | eqcomd 2170 | . . . . . 6 |
25 | 24 | feq2d 5319 | . . . . 5 |
26 | 14, 25 | mpbid 146 | . . . 4 |
27 | simpr3 994 | . . . . 5 | |
28 | 27 | feq1d 5318 | . . . 4 |
29 | 26, 28 | mpbird 166 | . . 3 |
30 | 27 | reseq1d 4877 | . . . . . 6 |
31 | ffn 5331 | . . . . . . . . . 10 | |
32 | fnresdisj 5292 | . . . . . . . . . 10 | |
33 | 1, 31, 32 | 3syl 17 | . . . . . . . . 9 |
34 | 12, 33 | mpbid 146 | . . . . . . . 8 |
35 | 34 | uneq1d 3270 | . . . . . . 7 |
36 | resundir 4892 | . . . . . . 7 | |
37 | uncom 3261 | . . . . . . . 8 | |
38 | un0 3437 | . . . . . . . 8 | |
39 | 37, 38 | eqtr2i 2186 | . . . . . . 7 |
40 | 35, 36, 39 | 3eqtr4g 2222 | . . . . . 6 |
41 | ffn 5331 | . . . . . . 7 | |
42 | fnresdm 5291 | . . . . . . 7 | |
43 | 10, 41, 42 | 3syl 17 | . . . . . 6 |
44 | 30, 40, 43 | 3eqtrd 2201 | . . . . 5 |
45 | 44 | fveq1d 5482 | . . . 4 |
46 | 16 | nn0zd 9302 | . . . . . 6 |
47 | 46 | peano2zd 9307 | . . . . 5 |
48 | snidg 3599 | . . . . 5 | |
49 | fvres 5504 | . . . . 5 | |
50 | 47, 48, 49 | 3syl 17 | . . . 4 |
51 | 5 | fveq1i 5481 | . . . . . 6 |
52 | fvsng 5675 | . . . . . 6 | |
53 | 51, 52 | syl5eq 2209 | . . . . 5 |
54 | 3, 4, 53 | syl2anc 409 | . . . 4 |
55 | 45, 50, 54 | 3eqtr3d 2205 | . . 3 |
56 | 27 | reseq1d 4877 | . . . 4 |
57 | incom 3309 | . . . . . . . 8 | |
58 | 57, 12 | syl5eq 2209 | . . . . . . 7 |
59 | ffn 5331 | . . . . . . . 8 | |
60 | fnresdisj 5292 | . . . . . . . 8 | |
61 | 8, 59, 60 | 3syl 17 | . . . . . . 7 |
62 | 58, 61 | mpbid 146 | . . . . . 6 |
63 | 62 | uneq2d 3271 | . . . . 5 |
64 | resundir 4892 | . . . . 5 | |
65 | un0 3437 | . . . . . 6 | |
66 | 65 | eqcomi 2168 | . . . . 5 |
67 | 63, 64, 66 | 3eqtr4g 2222 | . . . 4 |
68 | fnresdm 5291 | . . . . 5 | |
69 | 1, 31, 68 | 3syl 17 | . . . 4 |
70 | 56, 67, 69 | 3eqtrrd 2202 | . . 3 |
71 | 29, 55, 70 | 3jca 1166 | . 2 |
72 | simpr1 992 | . . . . 5 | |
73 | fzssp1 9992 | . . . . 5 | |
74 | fssres 5357 | . . . . 5 | |
75 | 72, 73, 74 | sylancl 410 | . . . 4 |
76 | simpr3 994 | . . . . 5 | |
77 | 76 | feq1d 5318 | . . . 4 |
78 | 75, 77 | mpbird 166 | . . 3 |
79 | simpr2 993 | . . . 4 | |
80 | 2 | adantr 274 | . . . . . . 7 |
81 | nnuz 9492 | . . . . . . 7 | |
82 | 80, 81 | eleqtrdi 2257 | . . . . . 6 |
83 | eluzfz2 9957 | . . . . . 6 | |
84 | 82, 83 | syl 14 | . . . . 5 |
85 | 72, 84 | ffvelrnd 5615 | . . . 4 |
86 | 79, 85 | eqeltrrd 2242 | . . 3 |
87 | ffn 5331 | . . . . . . . . 9 | |
88 | 72, 87 | syl 14 | . . . . . . . 8 |
89 | fnressn 5665 | . . . . . . . 8 | |
90 | 88, 84, 89 | syl2anc 409 | . . . . . . 7 |
91 | opeq2 3753 | . . . . . . . . 9 | |
92 | 91 | sneqd 3583 | . . . . . . . 8 |
93 | 79, 92 | syl 14 | . . . . . . 7 |
94 | 90, 93 | eqtrd 2197 | . . . . . 6 |
95 | 5, 94 | eqtr4id 2216 | . . . . 5 |
96 | 76, 95 | uneq12d 3272 | . . . 4 |
97 | simpl 108 | . . . . . . . 8 | |
98 | 97, 20 | eleqtrdi 2257 | . . . . . . 7 |
99 | 15, 98, 22 | sylancr 411 | . . . . . 6 |
100 | 99 | reseq2d 4878 | . . . . 5 |
101 | resundi 4891 | . . . . 5 | |
102 | 100, 101 | eqtr2di 2214 | . . . 4 |
103 | fnresdm 5291 | . . . . 5 | |
104 | 72, 87, 103 | 3syl 17 | . . . 4 |
105 | 96, 102, 104 | 3eqtrrd 2202 | . . 3 |
106 | 78, 86, 105 | 3jca 1166 | . 2 |
107 | 71, 106 | impbida 586 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 cun 3109 cin 3110 wss 3111 c0 3404 csn 3570 cop 3573 cres 4600 wfn 5177 wf 5178 cfv 5182 (class class class)co 5836 cc0 7744 c1 7745 caddc 7747 cmin 8060 cn 8848 cn0 9105 cz 9182 cuz 9457 cfz 9935 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: fseq1m1p1 10020 |
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