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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 12385 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpre 12385 | . . 3 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
3 | remulcl 10611 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2an 598 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ) |
5 | elrp 12379 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
6 | elrp 12379 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
7 | mulgt0 10707 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
8 | 5, 6, 7 | syl2anb 600 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 · 𝐵)) |
9 | elrp 12379 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℝ+ ↔ ((𝐴 · 𝐵) ∈ ℝ ∧ 0 < (𝐴 · 𝐵))) | |
10 | 4, 8, 9 | sylanbrc 586 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-mulrcl 10589 ax-rnegex 10597 ax-cnre 10599 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-rp 12378 |
This theorem is referenced by: rpmtmip 12401 rpmulcld 12435 moddi 13302 rpexpcl 13444 discr 13597 reccn2 14945 expcnv 15211 fprodrpcl 15302 rprisefaccl 15369 rpmsubg 20155 ovolscalem2 24118 aaliou3lem7 24945 aaliou3lem9 24946 cos02pilt1 25118 cosordlem 25122 logfac 25192 loglesqrt 25347 divsqrtsumlem 25565 basellem1 25666 pclogsum 25799 bclbnd 25864 bposlem7 25874 bposlem8 25875 bposlem9 25876 chebbnd1lem2 26054 dchrisum0lem3 26103 chpdifbndlem2 26138 pntrsumbnd2 26151 pntpbnd1a 26169 pntpbnd2 26171 pntibnd 26177 pntlemd 26178 pntlema 26180 pntlemb 26181 pntlemf 26189 pntlemo 26191 minvecolem3 28659 knoppndvlem18 33981 taupilem1 34735 taupilem2 34736 taupi 34737 ftc1anclem7 35136 ftc1anc 35138 isbnd2 35221 wallispilem4 42710 wallispi 42712 dirker2re 42734 dirkerdenne0 42735 dirkerper 42738 dirkertrigeq 42743 dirkercncflem2 42746 fourierdlem24 42773 sqwvfoura 42870 sqwvfourb 42871 amgmlemALT 45331 |
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