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Mirrors > Home > ILE Home > Th. List > recreclt | Unicode version |
Description: Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.) |
Ref | Expression |
---|---|
recreclt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recgt0 8608 | . . . 4 | |
2 | simpl 108 | . . . . . 6 | |
3 | gt0ap0 8388 | . . . . . 6 # | |
4 | 2, 3 | rerecclapd 8593 | . . . . 5 |
5 | 1re 7765 | . . . . 5 | |
6 | ltaddpos 8214 | . . . . 5 | |
7 | 4, 5, 6 | sylancl 409 | . . . 4 |
8 | 1, 7 | mpbid 146 | . . 3 |
9 | readdcl 7746 | . . . . 5 | |
10 | 5, 4, 9 | sylancr 410 | . . . 4 |
11 | 0lt1 7889 | . . . . . 6 | |
12 | 0re 7766 | . . . . . . . 8 | |
13 | lttr 7838 | . . . . . . . 8 | |
14 | 12, 5, 13 | mp3an12 1305 | . . . . . . 7 |
15 | 10, 14 | syl 14 | . . . . . 6 |
16 | 11, 15 | mpani 426 | . . . . 5 |
17 | 8, 16 | mpd 13 | . . . 4 |
18 | recgt1 8655 | . . . 4 | |
19 | 10, 17, 18 | syl2anc 408 | . . 3 |
20 | 8, 19 | mpbid 146 | . 2 |
21 | ltaddpos 8214 | . . . . . 6 | |
22 | 5, 4, 21 | sylancr 410 | . . . . 5 |
23 | 11, 22 | mpbii 147 | . . . 4 |
24 | 4 | recnd 7794 | . . . . 5 |
25 | ax-1cn 7713 | . . . . 5 | |
26 | addcom 7899 | . . . . 5 | |
27 | 24, 25, 26 | sylancl 409 | . . . 4 |
28 | 23, 27 | breqtrd 3954 | . . 3 |
29 | simpr 109 | . . . 4 | |
30 | ltrec1 8646 | . . . 4 | |
31 | 2, 29, 10, 17, 30 | syl22anc 1217 | . . 3 |
32 | 28, 31 | mpbid 146 | . 2 |
33 | 20, 32 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 clt 7800 cdiv 8432 |
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-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 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-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 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-reap 8337 df-ap 8344 df-div 8433 |
This theorem is referenced by: (None) |
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