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Mirrors > Home > MPE Home > Th. List > ltsubrp | Structured version Visualization version GIF version |
Description: Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.) |
Ref | Expression |
---|---|
ltsubrp | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrp 12245 | . 2 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
2 | ltsubpos 10986 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐵 ↔ (𝐴 − 𝐵) < 𝐴)) | |
3 | 2 | biimpd 230 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐵 → (𝐴 − 𝐵) < 𝐴)) |
4 | 3 | expcom 414 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 < 𝐵 → (𝐴 − 𝐵) < 𝐴))) |
5 | 4 | imp32 419 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 − 𝐵) < 𝐴) |
6 | 1, 5 | sylan2b 593 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2083 class class class wbr 4968 (class class class)co 7023 ℝcr 10389 0cc0 10390 < clt 10528 − cmin 10723 ℝ+crp 12243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-ltxr 10533 df-sub 10725 df-neg 10726 df-rp 12244 |
This theorem is referenced by: ltsubrpd 12317 bcval5 13532 climsup 14864 supcvg 15048 prmreclem6 16090 psgnunilem4 18360 iccntr 23116 reconnlem2 23122 opnreen 23126 ivthlem3 23741 leibpi 25206 chtppilim 25737 chto1ub 25738 dpmul4 30270 poimir 34477 ftc1anclem5 34523 stoweidlem59 41908 fouriersw 42080 |
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