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Mirrors > Home > MPE Home > Th. List > rpmulcl | Structured version Visualization version GIF version |
Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) |
Ref | Expression |
---|---|
rpmulcl | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 13040 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpre 13040 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
3 | remulcl 11237 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ) |
5 | elrp 13033 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
6 | elrp 13033 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
7 | mulgt0 11335 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
8 | 5, 6, 7 | syl2anb 598 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 · 𝐵)) |
9 | elrp 13033 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℝ+ ↔ ((𝐴 · 𝐵) ∈ ℝ ∧ 0 < (𝐴 · 𝐵))) | |
10 | 4, 8, 9 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 ℝcr 11151 0cc0 11152 · cmul 11157 < clt 11292 ℝ+crp 13031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-addrcl 11213 ax-mulrcl 11215 ax-rnegex 11223 ax-cnre 11225 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-rp 13032 |
This theorem is referenced by: rpmtmip 13056 rpmulcld 13090 moddi 13976 rpexpcl 14117 discr 14275 reccn2 15629 expcnv 15896 fprodrpcl 15988 rprisefaccl 16055 rpmsubg 21466 ovolscalem2 25562 aaliou3lem7 26405 aaliou3lem9 26406 cos02pilt1 26582 cosordlem 26586 logfac 26657 loglesqrt 26818 divsqrtsumlem 27037 basellem1 27138 pclogsum 27273 bclbnd 27338 bposlem7 27348 bposlem8 27349 bposlem9 27350 chebbnd1lem2 27528 dchrisum0lem3 27577 chpdifbndlem2 27612 pntrsumbnd2 27625 pntpbnd1a 27643 pntpbnd2 27645 pntibnd 27651 pntlemd 27652 pntlema 27654 pntlemb 27655 pntlemf 27663 pntlemo 27665 minvecolem3 30904 knoppndvlem18 36511 taupilem1 37303 taupilem2 37304 taupi 37305 ftc1anclem7 37685 ftc1anc 37687 isbnd2 37769 wallispilem4 46023 wallispi 46025 dirker2re 46047 dirkerdenne0 46048 dirkerper 46051 dirkertrigeq 46056 dirkercncflem2 46059 fourierdlem24 46086 sqwvfoura 46183 sqwvfourb 46184 amgmlemALT 49033 |
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