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Mirrors > Home > ILE Home > Th. List > ltaddpos | 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 7945 | . . 3 ⊢ 0 ∈ ℝ | |
2 | ltadd2 8363 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 + 0) < (𝐵 + 𝐴))) | |
3 | 1, 2 | mp3an1 1324 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 + 0) < (𝐵 + 𝐴))) |
4 | recn 7932 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
5 | 4 | addid1d 8093 | . . . 4 ⊢ (𝐵 ∈ ℝ → (𝐵 + 0) = 𝐵) |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 0) = 𝐵) |
7 | 6 | breq1d 4010 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 0) < (𝐵 + 𝐴) ↔ 𝐵 < (𝐵 + 𝐴))) |
8 | 3, 7 | bitrd 188 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5869 ℝcr 7798 0cc0 7799 + caddc 7802 < clt 7979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0id 7907 ax-rnegex 7908 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-iota 5174 df-fv 5220 df-ov 5872 df-pnf 7981 df-mnf 7982 df-ltxr 7984 |
This theorem is referenced by: ltaddpos2 8397 ltsubpos 8398 posdif 8399 ltaddposi 8441 ltaddposd 8473 ltp1 8787 recreclt 8843 ltaddrp 9675 ltoddhalfle 11878 |
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