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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 10400 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | |
2 | resubcl 10224 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
3 | 2 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
4 | elrp 11710 | . . . 4 ⊢ ((𝐵 − 𝐴) ∈ ℝ+ ↔ ((𝐵 − 𝐴) ∈ ℝ ∧ 0 < (𝐵 − 𝐴))) | |
5 | 4 | baib 942 | . . 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 270 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 < clt 9953 − cmin 10145 ℝ+crp 11708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-rp 11709 |
This theorem is referenced by: xralrple 11910 lincmb01cmp 12186 iccf1o 12187 expmulnbnd 12858 fsumlt 14373 expcnv 14435 blssps 22039 blss 22040 icchmeo 22548 icopnfcnv 22549 icopnfhmeo 22550 ivthlem2 23028 ivthlem3 23029 c1liplem1 23563 lhop1lem 23580 ftc1lem4 23606 aaliou3lem7 23908 abelthlem7 23996 cosordlem 24081 logdivlti 24170 cxpaddlelem 24292 atantan 24450 birthdaylem3 24480 lgamgulmlem2 24556 lgamgulmlem3 24557 chtppilimlem2 24963 pntrlog2bndlem5 25070 pntlemd 25083 pntlemc 25084 ostth2lem1 25107 ttgcontlem1 25565 lt2addrd 28903 signsplypnf 29953 knoppndvlem20 31692 ftc1cnnclem 32653 cvgdvgrat 37534 sge0gtfsumgt 39336 hoidmvlelem3 39487 vonioolem1 39571 smfmullem1 39676 smfmullem2 39677 smfmullem3 39678 |
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