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Mirrors > Home > ILE Home > Th. List > rpmulcld | GIF version |
Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
rpmulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | rpaddcld.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | rpmulcl 9466 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5774 · cmul 7625 ℝ+crp 9441 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-mulrcl 7719 ax-rnegex 7729 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-rp 9442 |
This theorem is referenced by: qbtwnrelemcalc 10033 cvg1nlemcxze 10754 cvg1nlemres 10757 resqrexlemnm 10790 resqrexlemcvg 10791 reccn2ap 11082 cvgratnnlembern 11292 cvgratnnlemrate 11299 cvgratnn 11300 eirraplem 11483 cosordlem 12930 |
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