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Mirrors > Home > ILE Home > Th. List > addgt0ii | GIF version |
Description: Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.) |
Ref | Expression |
---|---|
lt2.1 | ⊢ 𝐴 ∈ ℝ |
lt2.2 | ⊢ 𝐵 ∈ ℝ |
addgt0i.3 | ⊢ 0 < 𝐴 |
addgt0i.4 | ⊢ 0 < 𝐵 |
Ref | Expression |
---|---|
addgt0ii | ⊢ 0 < (𝐴 + 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addgt0i.3 | . 2 ⊢ 0 < 𝐴 | |
2 | addgt0i.4 | . 2 ⊢ 0 < 𝐵 | |
3 | lt2.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
4 | lt2.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
5 | 3, 4 | addgt0i 8020 | . 2 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵)) |
6 | 1, 2, 5 | mp2an 418 | 1 ⊢ 0 < (𝐴 + 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℝcr 7403 0cc0 7404 + caddc 7407 < clt 7576 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-pre-lttrn 7513 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4457 df-iota 4993 df-fv 5036 df-ov 5669 df-pnf 7578 df-mnf 7579 df-ltxr 7581 |
This theorem is referenced by: eqneg 8253 2pos 8567 3pos 8570 4pos 8573 5pos 8576 6pos 8577 7pos 8578 8pos 8579 9pos 8580 |
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