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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 12398 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpre 12398 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
3 | remulcl 10622 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ) |
5 | elrp 12392 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
6 | elrp 12392 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
7 | mulgt0 10718 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
8 | 5, 6, 7 | syl2anb 599 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 · 𝐵)) |
9 | elrp 12392 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℝ+ ↔ ((𝐴 · 𝐵) ∈ ℝ ∧ 0 < (𝐴 · 𝐵))) | |
10 | 4, 8, 9 | sylanbrc 585 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 · cmul 10542 < clt 10675 ℝ+crp 12390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-addrcl 10598 ax-mulrcl 10600 ax-rnegex 10608 ax-cnre 10610 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-rp 12391 |
This theorem is referenced by: rpmtmip 12414 rpmulcld 12448 moddi 13308 rpexpcl 13449 discr 13602 reccn2 14953 expcnv 15219 fprodrpcl 15310 rprisefaccl 15377 rpmsubg 20609 ovolscalem2 24115 aaliou3lem7 24938 aaliou3lem9 24939 cos02pilt1 25111 cosordlem 25115 logfac 25184 loglesqrt 25339 divsqrtsumlem 25557 basellem1 25658 pclogsum 25791 bclbnd 25856 bposlem7 25866 bposlem8 25867 bposlem9 25868 chebbnd1lem2 26046 dchrisum0lem3 26095 chpdifbndlem2 26130 pntrsumbnd2 26143 pntpbnd1a 26161 pntpbnd2 26163 pntibnd 26169 pntlemd 26170 pntlema 26172 pntlemb 26173 pntlemf 26181 pntlemo 26183 minvecolem3 28653 knoppndvlem18 33868 taupilem1 34605 taupilem2 34606 taupi 34607 ftc1anclem7 34988 ftc1anc 34990 isbnd2 35076 wallispilem4 42402 wallispi 42404 dirker2re 42426 dirkerdenne0 42427 dirkerper 42430 dirkertrigeq 42435 dirkercncflem2 42438 fourierdlem24 42465 sqwvfoura 42562 sqwvfourb 42563 amgmlemALT 44953 |
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