Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fprodge1 | Structured version Visualization version GIF version |
Description: If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodge1.ph | ⊢ Ⅎ𝑘𝜑 |
fprodge1.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodge1.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fprodge1.ge | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) |
Ref | Expression |
---|---|
fprodge1 | ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1xr 10703 | . 2 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 10698 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge1.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | 1re 10644 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | icossre 12820 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1[,)+∞) ⊆ ℝ) | |
6 | 4, 2, 5 | mp2an 690 | . . . . 5 ⊢ (1[,)+∞) ⊆ ℝ |
7 | ax-resscn 10597 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
8 | 6, 7 | sstri 3979 | . . . 4 ⊢ (1[,)+∞) ⊆ ℂ |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (1[,)+∞) ⊆ ℂ) |
10 | 1 | a1i 11 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ*) |
11 | 2 | a1i 11 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → +∞ ∈ ℝ*) |
12 | 6 | sseli 3966 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ) |
13 | 12 | adantr 483 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ) |
14 | 6 | sseli 3966 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ) |
15 | 14 | adantl 484 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
16 | 13, 15 | remulcld 10674 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ) |
17 | 16 | rexrd 10694 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ*) |
18 | 1t1e1 11802 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ) |
20 | 0le1 11166 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ 1) |
22 | icogelb 12791 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥) | |
23 | 1, 2, 22 | mp3an12 1447 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥) |
24 | 23 | adantr 483 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑥) |
25 | icogelb 12791 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) | |
26 | 1, 2, 25 | mp3an12 1447 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦) |
27 | 26 | adantl 484 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) |
28 | 19, 13, 19, 15, 21, 21, 24, 27 | lemul12ad 11585 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (1 · 1) ≤ (𝑥 · 𝑦)) |
29 | 18, 28 | eqbrtrrid 5105 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ (𝑥 · 𝑦)) |
30 | 16 | ltpnfd 12519 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) < +∞) |
31 | 10, 11, 17, 29, 30 | elicod 12790 | . . . 4 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
32 | 31 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞))) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
33 | fprodge1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
34 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℝ*) |
35 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → +∞ ∈ ℝ*) |
36 | fprodge1.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
37 | 36 | rexrd 10694 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
38 | fprodge1.ge | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) | |
39 | 36 | ltpnfd 12519 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < +∞) |
40 | 34, 35, 37, 38, 39 | elicod 12790 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (1[,)+∞)) |
41 | 1le1 11271 | . . . . 5 ⊢ 1 ≤ 1 | |
42 | ltpnf 12518 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
43 | 4, 42 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
44 | elico2 12803 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞))) | |
45 | 4, 2, 44 | mp2an 690 | . . . . 5 ⊢ (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞)) |
46 | 4, 41, 43, 45 | mpbir3an 1337 | . . . 4 ⊢ 1 ∈ (1[,)+∞) |
47 | 46 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (1[,)+∞)) |
48 | 3, 9, 32, 33, 40, 47 | fprodcllemf 15315 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) |
49 | icogelb 12791 | . 2 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
50 | 1, 2, 48, 49 | mp3an12i 1461 | 1 ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 Ⅎwnf 1783 ∈ wcel 2113 ⊆ wss 3939 class class class wbr 5069 (class class class)co 7159 Fincfn 8512 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 · cmul 10545 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 [,)cico 12743 ∏cprod 15262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-prod 15263 |
This theorem is referenced by: fprodle 15353 |
Copyright terms: Public domain | W3C validator |