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Mirrors > Home > MPE Home > Th. List > fprodge0 | Structured version Visualization version GIF version |
Description: If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodge0.kph | ⊢ Ⅎ𝑘𝜑 |
fprodge0.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodge0.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fprodge0.0leb | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
fprodge0 | ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10534 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfxr 10541 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge0.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | rge0ssre 12694 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
5 | ax-resscn 10440 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
6 | 4, 5 | sstri 3898 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
8 | ge0mulcl 12699 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 · 𝑦) ∈ (0[,)+∞)) | |
9 | 8 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 · 𝑦) ∈ (0[,)+∞)) |
10 | fprodge0.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
11 | fprodge0.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
12 | fprodge0.0leb | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
13 | elrege0 12692 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
14 | 11, 12, 13 | sylanbrc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
15 | 1re 10487 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | 0le1 11011 | . . . . 5 ⊢ 0 ≤ 1 | |
17 | ltpnf 12365 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
19 | 0re 10489 | . . . . . 6 ⊢ 0 ∈ ℝ | |
20 | elico2 12650 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞))) | |
21 | 19, 2, 20 | mp2an 688 | . . . . 5 ⊢ (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞)) |
22 | 15, 16, 18, 21 | mpbir3an 1334 | . . . 4 ⊢ 1 ∈ (0[,)+∞) |
23 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (0[,)+∞)) |
24 | 3, 7, 9, 10, 14, 23 | fprodcllemf 15145 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
25 | icogelb 12638 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
26 | 1, 2, 24, 25 | mp3an12i 1457 | 1 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 Ⅎwnf 1765 ∈ wcel 2081 ⊆ wss 3859 class class class wbr 4962 (class class class)co 7016 Fincfn 8357 ℂcc 10381 ℝcr 10382 0cc0 10383 1c1 10384 · cmul 10388 +∞cpnf 10518 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 [,)cico 12590 ∏cprod 15092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-ico 12594 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-prod 15093 |
This theorem is referenced by: fprodle 15183 hoiprodcl 42371 hoiprodcl3 42404 hoidmvcl 42406 hsphoidmvle2 42409 hsphoidmvle 42410 |
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