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Mirrors > Home > MPE Home > Th. List > ltaddpos | Structured version Visualization version GIF version |
Description: Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
ltaddpos | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10800 | . . 3 ⊢ 0 ∈ ℝ | |
2 | ltadd2 10901 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 + 0) < (𝐵 + 𝐴))) | |
3 | 1, 2 | mp3an1 1450 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 + 0) < (𝐵 + 𝐴))) |
4 | recn 10784 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | 4 | addid1d 10997 | . . . 4 ⊢ (𝐵 ∈ ℝ → (𝐵 + 0) = 𝐵) |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 0) = 𝐵) |
7 | 6 | breq1d 5049 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 0) < (𝐵 + 𝐴) ↔ 𝐵 < (𝐵 + 𝐴))) |
8 | 3, 7 | bitrd 282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 0cc0 10694 + caddc 10697 < clt 10832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 |
This theorem is referenced by: ltaddpos2 11288 ltsubpos 11289 posdif 11290 ltaddposi 11346 ltaddposd 11381 ltp1 11637 recreclt 11696 ltaddrp 12588 ccatval21sw 14107 ltoddhalfle 15885 dirkercncflem1 43262 |
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