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Mirrors > Home > MPE Home > Th. List > difrp | Structured version Visualization version GIF version |
Description: Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.) |
Ref | Expression |
---|---|
difrp | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ+)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | posdif 11754 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | |
2 | resubcl 11571 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
3 | 2 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
4 | elrp 13034 | . . . 4 ⊢ ((𝐵 − 𝐴) ∈ ℝ+ ↔ ((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴))) | |
5 | 4 | baib 535 | . . 3 ⊢ ((𝐵 − 𝐴) ∈ ℝ → ((𝐵 − 𝐴) ∈ ℝ+ ↔ 0 < (𝐵 − 𝐴))) |
6 | 3, 5 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝐴) ∈ ℝ+ ↔ 0 < (𝐵 − 𝐴))) |
7 | 1, 6 | bitr4d 282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 < clt 11293 − cmin 11490 ℝ+crp 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-rp 13033 |
This theorem is referenced by: xralrple 13244 lincmb01cmp 13532 iccf1o 13533 expmulnbnd 14271 fsumlt 15833 expcnv 15897 blssps 24450 blss 24451 icchmeo 24985 icchmeoOLD 24986 icopnfcnv 24987 icopnfhmeo 24988 ivthlem2 25501 ivthlem3 25502 c1liplem1 26050 lhop1lem 26067 ftc1lem4 26095 aaliou3lem7 26406 abelthlem7 26497 cosordlem 26587 logdivlti 26677 cxpaddlelem 26809 atantan 26981 birthdaylem3 27011 lgamgulmlem2 27088 lgamgulmlem3 27089 chtppilimlem2 27533 pntrlog2bndlem5 27640 pntlemd 27653 pntlemc 27654 ostth2lem1 27677 ttgcontlem1 28914 lt2addrd 32762 signsplypnf 34544 knoppndvlem20 36514 ftc1cnnclem 37678 fltnltalem 42649 fltnlta 42650 cvgdvgrat 44309 sge0gtfsumgt 46399 hoidmvlelem3 46553 vonioolem1 46636 smfmullem1 46747 smfmullem2 46748 smfmullem3 46749 |
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