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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 13036 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpre 13036 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
3 | remulcl 11243 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2an 594 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ) |
5 | elrp 13030 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
6 | elrp 13030 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
7 | mulgt0 11341 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
8 | 5, 6, 7 | syl2anb 596 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 · 𝐵)) |
9 | elrp 13030 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℝ+ ↔ ((𝐴 · 𝐵) ∈ ℝ ∧ 0 < (𝐴 · 𝐵))) | |
10 | 4, 8, 9 | sylanbrc 581 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝcr 11157 0cc0 11158 · cmul 11163 < clt 11298 ℝ+crp 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-addrcl 11219 ax-mulrcl 11221 ax-rnegex 11229 ax-cnre 11231 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-rp 13029 |
This theorem is referenced by: rpmtmip 13052 rpmulcld 13086 moddi 13959 rpexpcl 14100 discr 14257 reccn2 15599 expcnv 15868 fprodrpcl 15958 rprisefaccl 16025 rpmsubg 21428 ovolscalem2 25534 aaliou3lem7 26377 aaliou3lem9 26378 cos02pilt1 26553 cosordlem 26557 logfac 26628 loglesqrt 26789 divsqrtsumlem 27008 basellem1 27109 pclogsum 27244 bclbnd 27309 bposlem7 27319 bposlem8 27320 bposlem9 27321 chebbnd1lem2 27499 dchrisum0lem3 27548 chpdifbndlem2 27583 pntrsumbnd2 27596 pntpbnd1a 27614 pntpbnd2 27616 pntibnd 27622 pntlemd 27623 pntlema 27625 pntlemb 27626 pntlemf 27634 pntlemo 27636 minvecolem3 30809 knoppndvlem18 36232 taupilem1 37028 taupilem2 37029 taupi 37030 ftc1anclem7 37400 ftc1anc 37402 isbnd2 37484 wallispilem4 45689 wallispi 45691 dirker2re 45713 dirkerdenne0 45714 dirkerper 45717 dirkertrigeq 45722 dirkercncflem2 45725 fourierdlem24 45752 sqwvfoura 45849 sqwvfourb 45850 amgmlemALT 48551 |
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