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Mirrors > Home > ILE Home > Th. List > icoshftf1o | Unicode version |
Description: Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
icoshftf1o.1 |
Ref | Expression |
---|---|
icoshftf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoshft 9766 | . . 3 | |
2 | 1 | ralrimiv 2502 | . 2 |
3 | readdcl 7739 | . . . . . . . . 9 | |
4 | 3 | 3adant2 1000 | . . . . . . . 8 |
5 | readdcl 7739 | . . . . . . . . 9 | |
6 | 5 | 3adant1 999 | . . . . . . . 8 |
7 | renegcl 8016 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant3 1004 | . . . . . . . 8 |
9 | icoshft 9766 | . . . . . . . 8 | |
10 | 4, 6, 8, 9 | syl3anc 1216 | . . . . . . 7 |
11 | 10 | imp 123 | . . . . . 6 |
12 | 6 | rexrd 7808 | . . . . . . . . . 10 |
13 | icossre 9730 | . . . . . . . . . 10 | |
14 | 4, 12, 13 | syl2anc 408 | . . . . . . . . 9 |
15 | 14 | sselda 3092 | . . . . . . . 8 |
16 | 15 | recnd 7787 | . . . . . . 7 |
17 | simpl3 986 | . . . . . . . 8 | |
18 | 17 | recnd 7787 | . . . . . . 7 |
19 | 16, 18 | negsubd 8072 | . . . . . 6 |
20 | 4 | recnd 7787 | . . . . . . . . . 10 |
21 | simp3 983 | . . . . . . . . . . 11 | |
22 | 21 | recnd 7787 | . . . . . . . . . 10 |
23 | 20, 22 | negsubd 8072 | . . . . . . . . 9 |
24 | simp1 981 | . . . . . . . . . . 11 | |
25 | 24 | recnd 7787 | . . . . . . . . . 10 |
26 | 25, 22 | pncand 8067 | . . . . . . . . 9 |
27 | 23, 26 | eqtrd 2170 | . . . . . . . 8 |
28 | 6 | recnd 7787 | . . . . . . . . . 10 |
29 | 28, 22 | negsubd 8072 | . . . . . . . . 9 |
30 | simp2 982 | . . . . . . . . . . 11 | |
31 | 30 | recnd 7787 | . . . . . . . . . 10 |
32 | 31, 22 | pncand 8067 | . . . . . . . . 9 |
33 | 29, 32 | eqtrd 2170 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 5785 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 11, 19, 35 | 3eltr3d 2220 | . . . . 5 |
37 | reueq 2878 | . . . . 5 | |
38 | 36, 37 | sylib 121 | . . . 4 |
39 | 15 | adantr 274 | . . . . . . . 8 |
40 | 39 | recnd 7787 | . . . . . . 7 |
41 | simpll3 1022 | . . . . . . . 8 | |
42 | 41 | recnd 7787 | . . . . . . 7 |
43 | simpl1 984 | . . . . . . . . . 10 | |
44 | simpl2 985 | . . . . . . . . . . 11 | |
45 | 44 | rexrd 7808 | . . . . . . . . . 10 |
46 | icossre 9730 | . . . . . . . . . 10 | |
47 | 43, 45, 46 | syl2anc 408 | . . . . . . . . 9 |
48 | 47 | sselda 3092 | . . . . . . . 8 |
49 | 48 | recnd 7787 | . . . . . . 7 |
50 | 40, 42, 49 | subadd2d 8085 | . . . . . 6 |
51 | eqcom 2139 | . . . . . 6 | |
52 | eqcom 2139 | . . . . . 6 | |
53 | 50, 51, 52 | 3bitr4g 222 | . . . . 5 |
54 | 53 | reubidva 2611 | . . . 4 |
55 | 38, 54 | mpbid 146 | . . 3 |
56 | 55 | ralrimiva 2503 | . 2 |
57 | icoshftf1o.1 | . . 3 | |
58 | 57 | f1ompt 5564 | . 2 |
59 | 2, 56, 58 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2414 wreu 2416 wss 3066 cmpt 3984 wf1o 5117 (class class class)co 5767 cr 7612 caddc 7616 cxr 7792 cmin 7926 cneg 7927 cico 9666 |
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-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 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-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 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-ico 9670 |
This theorem is referenced by: (None) |
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