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Mirrors > Home > ILE Home > Th. List > mulgt0i | GIF version |
Description: The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
mulgt0i | ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | lt.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | axmulgt0 7860 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 · cmul 7649 < clt 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-mulrcl 7743 ax-rnegex 7753 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 |
This theorem is referenced by: mulgt0ii 7898 |
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