Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | fprodge0.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | elrege0 12149 | . . . . . . 7 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) | |
3 | 2 | simplbi 475 | . . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℝ) |
4 | 3 | ssriv 3572 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ |
5 | ax-resscn 9872 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
6 | 4, 5 | sstri 3577 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
8 | ge0mulcl 12156 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 · 𝑦) ∈ (0[,)+∞)) | |
9 | 8 | adantl 481 | . . 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 12149 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
14 | 11, 12, 13 | sylanbrc 695 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
15 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
16 | 0le1 10430 | . . . . . 6 ⊢ 0 ≤ 1 | |
17 | ltpnf 11830 | . . . . . . 7 ⊢ (1 ∈ ℝ → 1 < +∞) | |
18 | 15, 17 | ax-mp 5 | . . . . . 6 ⊢ 1 < +∞ |
19 | 15, 16, 18 | 3pm3.2i 1232 | . . . . 5 ⊢ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞) |
20 | 0e0icopnf 12153 | . . . . . . 7 ⊢ 0 ∈ (0[,)+∞) | |
21 | 4, 20 | sselii 3565 | . . . . . 6 ⊢ 0 ∈ ℝ |
22 | pnfxr 9971 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
23 | elico2 12108 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞))) | |
24 | 21, 22, 23 | mp2an 704 | . . . . 5 ⊢ (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞)) |
25 | 19, 24 | mpbir 220 | . . . 4 ⊢ 1 ∈ (0[,)+∞) |
26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (0[,)+∞)) |
27 | 1, 7, 9, 10, 14, 26 | fprodcllemf 14527 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
28 | 0xr 9965 | . . . 4 ⊢ 0 ∈ ℝ* | |
29 | 28 | a1i 11 | . . 3 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) → 0 ∈ ℝ*) |
30 | 22 | a1i 11 | . . 3 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) → +∞ ∈ ℝ*) |
31 | id 22 | . . 3 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) → ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) | |
32 | icogelb 12096 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
33 | 29, 30, 31, 32 | syl3anc 1318 | . 2 ⊢ (∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
34 | 27, 33 | syl 17 | 1 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 Ⅎwnf 1699 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,)cico 12048 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 |
This theorem is referenced by: fprodle 14566 hoiprodcl 39437 hoiprodcl3 39470 hoidmvcl 39472 hsphoidmvle2 39475 hsphoidmvle 39476 |
Copyright terms: Public domain | W3C validator |