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Mirrors > Home > MPE Home > Th. List > icoshftf1o | Structured version Visualization version 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 12294 | . . 3 | |
2 | 1 | ralrimiv 2965 | . 2 |
3 | readdcl 10019 | . . . . . . . . 9 | |
4 | 3 | 3adant2 1080 | . . . . . . . 8 |
5 | readdcl 10019 | . . . . . . . . 9 | |
6 | 5 | 3adant1 1079 | . . . . . . . 8 |
7 | renegcl 10344 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant3 1084 | . . . . . . . 8 |
9 | icoshft 12294 | . . . . . . . 8 | |
10 | 4, 6, 8, 9 | syl3anc 1326 | . . . . . . 7 |
11 | 10 | imp 445 | . . . . . 6 |
12 | 6 | rexrd 10089 | . . . . . . . . . 10 |
13 | icossre 12254 | . . . . . . . . . 10 | |
14 | 4, 12, 13 | syl2anc 693 | . . . . . . . . 9 |
15 | 14 | sselda 3603 | . . . . . . . 8 |
16 | 15 | recnd 10068 | . . . . . . 7 |
17 | simpl3 1066 | . . . . . . . 8 | |
18 | 17 | recnd 10068 | . . . . . . 7 |
19 | 16, 18 | negsubd 10398 | . . . . . 6 |
20 | 4 | recnd 10068 | . . . . . . . . . 10 |
21 | simp3 1063 | . . . . . . . . . . 11 | |
22 | 21 | recnd 10068 | . . . . . . . . . 10 |
23 | 20, 22 | negsubd 10398 | . . . . . . . . 9 |
24 | simp1 1061 | . . . . . . . . . . 11 | |
25 | 24 | recnd 10068 | . . . . . . . . . 10 |
26 | 25, 22 | pncand 10393 | . . . . . . . . 9 |
27 | 23, 26 | eqtrd 2656 | . . . . . . . 8 |
28 | 6 | recnd 10068 | . . . . . . . . . 10 |
29 | 28, 22 | negsubd 10398 | . . . . . . . . 9 |
30 | simp2 1062 | . . . . . . . . . . 11 | |
31 | 30 | recnd 10068 | . . . . . . . . . 10 |
32 | 31, 22 | pncand 10393 | . . . . . . . . 9 |
33 | 29, 32 | eqtrd 2656 | . . . . . . . 8 |
34 | 27, 33 | oveq12d 6668 | . . . . . . 7 |
35 | 34 | adantr 481 | . . . . . 6 |
36 | 11, 19, 35 | 3eltr3d 2715 | . . . . 5 |
37 | reueq 3404 | . . . . 5 | |
38 | 36, 37 | sylib 208 | . . . 4 |
39 | 15 | adantr 481 | . . . . . . . 8 |
40 | 39 | recnd 10068 | . . . . . . 7 |
41 | simpll3 1102 | . . . . . . . 8 | |
42 | 41 | recnd 10068 | . . . . . . 7 |
43 | simpl1 1064 | . . . . . . . . . 10 | |
44 | simpl2 1065 | . . . . . . . . . . 11 | |
45 | 44 | rexrd 10089 | . . . . . . . . . 10 |
46 | icossre 12254 | . . . . . . . . . 10 | |
47 | 43, 45, 46 | syl2anc 693 | . . . . . . . . 9 |
48 | 47 | sselda 3603 | . . . . . . . 8 |
49 | 48 | recnd 10068 | . . . . . . 7 |
50 | 40, 42, 49 | subadd2d 10411 | . . . . . 6 |
51 | eqcom 2629 | . . . . . 6 | |
52 | eqcom 2629 | . . . . . 6 | |
53 | 50, 51, 52 | 3bitr4g 303 | . . . . 5 |
54 | 53 | reubidva 3125 | . . . 4 |
55 | 38, 54 | mpbid 222 | . . 3 |
56 | 55 | ralrimiva 2966 | . 2 |
57 | icoshftf1o.1 | . . 3 | |
58 | 57 | f1ompt 6382 | . 2 |
59 | 2, 56, 58 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wreu 2914 wss 3574 cmpt 4729 wf1o 5887 (class class class)co 6650 cr 9935 caddc 9939 cxr 10073 cmin 10266 cneg 10267 cico 12177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-ico 12181 |
This theorem is referenced by: (None) |
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