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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 13043 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
| 2 | rpre 13043 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
| 3 | remulcl 11240 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ) |
| 5 | elrp 13036 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
| 6 | elrp 13036 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
| 7 | mulgt0 11338 | . . 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 13036 | . 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 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 · cmul 11160 < clt 11295 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-mulrcl 11218 ax-rnegex 11226 ax-cnre 11228 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-rp 13035 |
| This theorem is referenced by: rpmtmip 13059 rpmulcld 13093 moddi 13980 rpexpcl 14121 discr 14279 reccn2 15633 expcnv 15900 fprodrpcl 15992 rprisefaccl 16059 rpmsubg 21449 ovolscalem2 25549 aaliou3lem7 26391 aaliou3lem9 26392 cos02pilt1 26568 cosordlem 26572 logfac 26643 loglesqrt 26804 divsqrtsumlem 27023 basellem1 27124 pclogsum 27259 bclbnd 27324 bposlem7 27334 bposlem8 27335 bposlem9 27336 chebbnd1lem2 27514 dchrisum0lem3 27563 chpdifbndlem2 27598 pntrsumbnd2 27611 pntpbnd1a 27629 pntpbnd2 27631 pntibnd 27637 pntlemd 27638 pntlema 27640 pntlemb 27641 pntlemf 27649 pntlemo 27651 minvecolem3 30895 knoppndvlem18 36530 taupilem1 37322 taupilem2 37323 taupi 37324 ftc1anclem7 37706 ftc1anc 37708 isbnd2 37790 wallispilem4 46083 wallispi 46085 dirker2re 46107 dirkerdenne0 46108 dirkerper 46111 dirkertrigeq 46116 dirkercncflem2 46119 fourierdlem24 46146 sqwvfoura 46243 sqwvfourb 46244 amgmlemALT 49322 |
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