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Mirrors > Home > MPE Home > Th. List > icoshft | Structured version Visualization version Unicode version |
Description: A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.) |
Ref | Expression |
---|---|
icoshft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10085 | . . . . . 6 | |
2 | elico2 12237 | . . . . . 6 | |
3 | 1, 2 | sylan2 491 | . . . . 5 |
4 | 3 | biimpd 219 | . . . 4 |
5 | 4 | 3adant3 1081 | . . 3 |
6 | 3anass 1042 | . . 3 | |
7 | 5, 6 | syl6ib 241 | . 2 |
8 | leadd1 10496 | . . . . . . . . . 10 | |
9 | 8 | 3com12 1269 | . . . . . . . . 9 |
10 | 9 | 3expib 1268 | . . . . . . . 8 |
11 | 10 | com12 32 | . . . . . . 7 |
12 | 11 | 3adant2 1080 | . . . . . 6 |
13 | 12 | imp 445 | . . . . 5 |
14 | ltadd1 10495 | . . . . . . . . 9 | |
15 | 14 | 3expib 1268 | . . . . . . . 8 |
16 | 15 | com12 32 | . . . . . . 7 |
17 | 16 | 3adant1 1079 | . . . . . 6 |
18 | 17 | imp 445 | . . . . 5 |
19 | 13, 18 | anbi12d 747 | . . . 4 |
20 | 19 | pm5.32da 673 | . . 3 |
21 | readdcl 10019 | . . . . . . . 8 | |
22 | 21 | expcom 451 | . . . . . . 7 |
23 | 22 | anim1d 588 | . . . . . 6 |
24 | 3anass 1042 | . . . . . 6 | |
25 | 23, 24 | syl6ibr 242 | . . . . 5 |
26 | 25 | 3ad2ant3 1084 | . . . 4 |
27 | readdcl 10019 | . . . . . 6 | |
28 | 27 | 3adant2 1080 | . . . . 5 |
29 | readdcl 10019 | . . . . . 6 | |
30 | 29 | 3adant1 1079 | . . . . 5 |
31 | rexr 10085 | . . . . . . 7 | |
32 | elico2 12237 | . . . . . . 7 | |
33 | 31, 32 | sylan2 491 | . . . . . 6 |
34 | 33 | biimprd 238 | . . . . 5 |
35 | 28, 30, 34 | syl2anc 693 | . . . 4 |
36 | 26, 35 | syld 47 | . . 3 |
37 | 20, 36 | sylbid 230 | . 2 |
38 | 7, 37 | syld 47 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 caddc 9939 cxr 10073 clt 10074 cle 10075 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-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-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-ico 12181 |
This theorem is referenced by: icoshftf1o 12295 |
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