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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 9728 | . . 3 | |
2 | 1 | ralrimiv 2481 | . 2 |
3 | readdcl 7714 | . . . . . . . . 9 | |
4 | 3 | 3adant2 985 | . . . . . . . 8 |
5 | readdcl 7714 | . . . . . . . . 9 | |
6 | 5 | 3adant1 984 | . . . . . . . 8 |
7 | renegcl 7991 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant3 989 | . . . . . . . 8 |
9 | icoshft 9728 | . . . . . . . 8 | |
10 | 4, 6, 8, 9 | syl3anc 1201 | . . . . . . 7 |
11 | 10 | imp 123 | . . . . . 6 |
12 | 6 | rexrd 7783 | . . . . . . . . . 10 |
13 | icossre 9692 | . . . . . . . . . 10 | |
14 | 4, 12, 13 | syl2anc 408 | . . . . . . . . 9 |
15 | 14 | sselda 3067 | . . . . . . . 8 |
16 | 15 | recnd 7762 | . . . . . . 7 |
17 | simpl3 971 | . . . . . . . 8 | |
18 | 17 | recnd 7762 | . . . . . . 7 |
19 | 16, 18 | negsubd 8047 | . . . . . 6 |
20 | 4 | recnd 7762 | . . . . . . . . . 10 |
21 | simp3 968 | . . . . . . . . . . 11 | |
22 | 21 | recnd 7762 | . . . . . . . . . 10 |
23 | 20, 22 | negsubd 8047 | . . . . . . . . 9 |
24 | simp1 966 | . . . . . . . . . . 11 | |
25 | 24 | recnd 7762 | . . . . . . . . . 10 |
26 | 25, 22 | pncand 8042 | . . . . . . . . 9 |
27 | 23, 26 | eqtrd 2150 | . . . . . . . 8 |
28 | 6 | recnd 7762 | . . . . . . . . . 10 |
29 | 28, 22 | negsubd 8047 | . . . . . . . . 9 |
30 | simp2 967 | . . . . . . . . . . 11 | |
31 | 30 | recnd 7762 | . . . . . . . . . 10 |
32 | 31, 22 | pncand 8042 | . . . . . . . . 9 |
33 | 29, 32 | eqtrd 2150 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 5760 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 11, 19, 35 | 3eltr3d 2200 | . . . . 5 |
37 | reueq 2856 | . . . . 5 | |
38 | 36, 37 | sylib 121 | . . . 4 |
39 | 15 | adantr 274 | . . . . . . . 8 |
40 | 39 | recnd 7762 | . . . . . . 7 |
41 | simpll3 1007 | . . . . . . . 8 | |
42 | 41 | recnd 7762 | . . . . . . 7 |
43 | simpl1 969 | . . . . . . . . . 10 | |
44 | simpl2 970 | . . . . . . . . . . 11 | |
45 | 44 | rexrd 7783 | . . . . . . . . . 10 |
46 | icossre 9692 | . . . . . . . . . 10 | |
47 | 43, 45, 46 | syl2anc 408 | . . . . . . . . 9 |
48 | 47 | sselda 3067 | . . . . . . . 8 |
49 | 48 | recnd 7762 | . . . . . . 7 |
50 | 40, 42, 49 | subadd2d 8060 | . . . . . 6 |
51 | eqcom 2119 | . . . . . 6 | |
52 | eqcom 2119 | . . . . . 6 | |
53 | 50, 51, 52 | 3bitr4g 222 | . . . . 5 |
54 | 53 | reubidva 2590 | . . . 4 |
55 | 38, 54 | mpbid 146 | . . 3 |
56 | 55 | ralrimiva 2482 | . 2 |
57 | icoshftf1o.1 | . . 3 | |
58 | 57 | f1ompt 5539 | . 2 |
59 | 2, 56, 58 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wral 2393 wreu 2395 wss 3041 cmpt 3959 wf1o 5092 (class class class)co 5742 cr 7587 caddc 7591 cxr 7767 cmin 7901 cneg 7902 cico 9628 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-ico 9632 |
This theorem is referenced by: (None) |
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