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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 9773 | . . 3 | |
2 | 1 | ralrimiv 2504 | . 2 |
3 | readdcl 7746 | . . . . . . . . 9 | |
4 | 3 | 3adant2 1000 | . . . . . . . 8 |
5 | readdcl 7746 | . . . . . . . . 9 | |
6 | 5 | 3adant1 999 | . . . . . . . 8 |
7 | renegcl 8023 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant3 1004 | . . . . . . . 8 |
9 | icoshft 9773 | . . . . . . . 8 | |
10 | 4, 6, 8, 9 | syl3anc 1216 | . . . . . . 7 |
11 | 10 | imp 123 | . . . . . 6 |
12 | 6 | rexrd 7815 | . . . . . . . . . 10 |
13 | icossre 9737 | . . . . . . . . . 10 | |
14 | 4, 12, 13 | syl2anc 408 | . . . . . . . . 9 |
15 | 14 | sselda 3097 | . . . . . . . 8 |
16 | 15 | recnd 7794 | . . . . . . 7 |
17 | simpl3 986 | . . . . . . . 8 | |
18 | 17 | recnd 7794 | . . . . . . 7 |
19 | 16, 18 | negsubd 8079 | . . . . . 6 |
20 | 4 | recnd 7794 | . . . . . . . . . 10 |
21 | simp3 983 | . . . . . . . . . . 11 | |
22 | 21 | recnd 7794 | . . . . . . . . . 10 |
23 | 20, 22 | negsubd 8079 | . . . . . . . . 9 |
24 | simp1 981 | . . . . . . . . . . 11 | |
25 | 24 | recnd 7794 | . . . . . . . . . 10 |
26 | 25, 22 | pncand 8074 | . . . . . . . . 9 |
27 | 23, 26 | eqtrd 2172 | . . . . . . . 8 |
28 | 6 | recnd 7794 | . . . . . . . . . 10 |
29 | 28, 22 | negsubd 8079 | . . . . . . . . 9 |
30 | simp2 982 | . . . . . . . . . . 11 | |
31 | 30 | recnd 7794 | . . . . . . . . . 10 |
32 | 31, 22 | pncand 8074 | . . . . . . . . 9 |
33 | 29, 32 | eqtrd 2172 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 5792 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 11, 19, 35 | 3eltr3d 2222 | . . . . 5 |
37 | reueq 2883 | . . . . 5 | |
38 | 36, 37 | sylib 121 | . . . 4 |
39 | 15 | adantr 274 | . . . . . . . 8 |
40 | 39 | recnd 7794 | . . . . . . 7 |
41 | simpll3 1022 | . . . . . . . 8 | |
42 | 41 | recnd 7794 | . . . . . . 7 |
43 | simpl1 984 | . . . . . . . . . 10 | |
44 | simpl2 985 | . . . . . . . . . . 11 | |
45 | 44 | rexrd 7815 | . . . . . . . . . 10 |
46 | icossre 9737 | . . . . . . . . . 10 | |
47 | 43, 45, 46 | syl2anc 408 | . . . . . . . . 9 |
48 | 47 | sselda 3097 | . . . . . . . 8 |
49 | 48 | recnd 7794 | . . . . . . 7 |
50 | 40, 42, 49 | subadd2d 8092 | . . . . . 6 |
51 | eqcom 2141 | . . . . . 6 | |
52 | eqcom 2141 | . . . . . 6 | |
53 | 50, 51, 52 | 3bitr4g 222 | . . . . 5 |
54 | 53 | reubidva 2613 | . . . 4 |
55 | 38, 54 | mpbid 146 | . . 3 |
56 | 55 | ralrimiva 2505 | . 2 |
57 | icoshftf1o.1 | . . 3 | |
58 | 57 | f1ompt 5571 | . 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 2416 wreu 2418 wss 3071 cmpt 3989 wf1o 5122 (class class class)co 5774 cr 7619 caddc 7623 cxr 7799 cmin 7933 cneg 7934 cico 9673 |
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-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 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-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 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-ico 9677 |
This theorem is referenced by: (None) |
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