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Mirrors > Home > MPE Home > Th. List > Mathboxes > lt4addmuld | Structured version Visualization version Unicode version |
Description: If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lt4addmuld.a | |
lt4addmuld.b | |
lt4addmuld.c | |
lt4addmuld.d | |
lt4addmuld.e | |
lt4addmuld.alte | |
lt4addmuld.blte | |
lt4addmuld.clte | |
lt4addmuld.dlte |
Ref | Expression |
---|---|
lt4addmuld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt4addmuld.a | . . . . 5 | |
2 | lt4addmuld.b | . . . . 5 | |
3 | 1, 2 | readdcld 10069 | . . . 4 |
4 | lt4addmuld.c | . . . 4 | |
5 | 3, 4 | readdcld 10069 | . . 3 |
6 | lt4addmuld.d | . . 3 | |
7 | 3re 11094 | . . . . 5 | |
8 | 7 | a1i 11 | . . . 4 |
9 | lt4addmuld.e | . . . 4 | |
10 | 8, 9 | remulcld 10070 | . . 3 |
11 | lt4addmuld.alte | . . . 4 | |
12 | lt4addmuld.blte | . . . 4 | |
13 | lt4addmuld.clte | . . . 4 | |
14 | 1, 2, 4, 9, 11, 12, 13 | lt3addmuld 39515 | . . 3 |
15 | lt4addmuld.dlte | . . 3 | |
16 | 5, 6, 10, 9, 14, 15 | lt2addd 10650 | . 2 |
17 | df-4 11081 | . . . . 5 | |
18 | 17 | a1i 11 | . . . 4 |
19 | 18 | oveq1d 6665 | . . 3 |
20 | 8 | recnd 10068 | . . . 4 |
21 | 9 | recnd 10068 | . . . 4 |
22 | 20, 21 | adddirp1d 10066 | . . 3 |
23 | 19, 22 | eqtr2d 2657 | . 2 |
24 | 16, 23 | breqtrd 4679 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 c1 9937 caddc 9939 cmul 9941 clt 10074 c3 11071 c4 11072 |
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-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-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-2 11079 df-3 11080 df-4 11081 |
This theorem is referenced by: limclner 39883 |
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