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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 10547 | . 2 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 10541 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge1.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | 1re 10487 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | icossre 12667 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1[,)+∞) ⊆ ℝ) | |
6 | 4, 2, 5 | mp2an 688 | . . . . 5 ⊢ (1[,)+∞) ⊆ ℝ |
7 | ax-resscn 10440 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
8 | 6, 7 | sstri 3898 | . . . 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 3885 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ) |
13 | 12 | adantr 481 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ) |
14 | 6 | sseli 3885 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ) |
15 | 14 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
16 | 13, 15 | remulcld 10517 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ) |
17 | 16 | rexrd 10537 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ*) |
18 | 1t1e1 11647 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ) |
20 | 0le1 11011 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ 1) |
22 | icogelb 12638 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥) | |
23 | 1, 2, 22 | mp3an12 1443 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥) |
24 | 23 | adantr 481 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑥) |
25 | icogelb 12638 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) | |
26 | 1, 2, 25 | mp3an12 1443 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦) |
27 | 26 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) |
28 | 19, 13, 19, 15, 21, 21, 24, 27 | lemul12ad 11430 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (1 · 1) ≤ (𝑥 · 𝑦)) |
29 | 18, 28 | eqbrtrrid 4998 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ (𝑥 · 𝑦)) |
30 | 16 | ltpnfd 12366 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) < +∞) |
31 | 10, 11, 17, 29, 30 | elicod 12637 | . . . 4 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
32 | 31 | adantl 482 | . . 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 10537 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
38 | fprodge1.ge | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) | |
39 | 36 | ltpnfd 12366 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < +∞) |
40 | 34, 35, 37, 38, 39 | elicod 12637 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (1[,)+∞)) |
41 | 1le1 11116 | . . . . 5 ⊢ 1 ≤ 1 | |
42 | ltpnf 12365 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
43 | 4, 42 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
44 | elico2 12650 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞))) | |
45 | 4, 2, 44 | mp2an 688 | . . . . 5 ⊢ (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞)) |
46 | 4, 41, 43, 45 | mpbir3an 1334 | . . . 4 ⊢ 1 ∈ (1[,)+∞) |
47 | 46 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (1[,)+∞)) |
48 | 3, 9, 32, 33, 40, 47 | fprodcllemf 15145 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) |
49 | icogelb 12638 | . 2 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
50 | 1, 2, 48, 49 | mp3an12i 1457 | 1 ⊢ (𝜑 → 1 ≤ ∏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 |
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