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Mirrors > Home > MPE Home > Th. List > lt2addmuld | Structured version Visualization version GIF version |
Description: If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lt2addmuld.a | โข (๐ โ ๐ด โ โ) |
lt2addmuld.b | โข (๐ โ ๐ต โ โ) |
lt2addmuld.c | โข (๐ โ ๐ถ โ โ) |
lt2addmuld.altc | โข (๐ โ ๐ด < ๐ถ) |
lt2addmuld.bltc | โข (๐ โ ๐ต < ๐ถ) |
Ref | Expression |
---|---|
lt2addmuld | โข (๐ โ (๐ด + ๐ต) < (2 ยท ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2addmuld.a | . . 3 โข (๐ โ ๐ด โ โ) | |
2 | lt2addmuld.b | . . 3 โข (๐ โ ๐ต โ โ) | |
3 | lt2addmuld.c | . . 3 โข (๐ โ ๐ถ โ โ) | |
4 | lt2addmuld.altc | . . 3 โข (๐ โ ๐ด < ๐ถ) | |
5 | lt2addmuld.bltc | . . 3 โข (๐ โ ๐ต < ๐ถ) | |
6 | 1, 2, 3, 3, 4, 5 | lt2addd 11779 | . 2 โข (๐ โ (๐ด + ๐ต) < (๐ถ + ๐ถ)) |
7 | 3 | recnd 11184 | . . 3 โข (๐ โ ๐ถ โ โ) |
8 | 7 | 2timesd 12397 | . 2 โข (๐ โ (2 ยท ๐ถ) = (๐ถ + ๐ถ)) |
9 | 6, 8 | breqtrrd 5134 | 1 โข (๐ โ (๐ด + ๐ต) < (2 ยท ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2107 class class class wbr 5106 (class class class)co 7358 โcr 11051 + caddc 11055 ยท cmul 11057 < clt 11190 2c2 12209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-2 12217 |
This theorem is referenced by: crctcshwlkn0lem5 28762 rmspecsqrtnq 41232 lt3addmuld 43542 |
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