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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 11317 | . 2 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 11312 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge1.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | 1re 11258 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | icossre 13464 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1[,)+∞) ⊆ ℝ) | |
6 | 4, 2, 5 | mp2an 692 | . . . . 5 ⊢ (1[,)+∞) ⊆ ℝ |
7 | ax-resscn 11209 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
8 | 6, 7 | sstri 4004 | . . . 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 3990 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ) |
13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ) |
14 | 6 | sseli 3990 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ) |
15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
16 | 13, 15 | remulcld 11288 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ) |
17 | 16 | rexrd 11308 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ*) |
18 | 1t1e1 12425 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ) |
20 | 0le1 11783 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ 1) |
22 | icogelb 13434 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥) | |
23 | 1, 2, 22 | mp3an12 1450 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥) |
24 | 23 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑥) |
25 | icogelb 13434 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) | |
26 | 1, 2, 25 | mp3an12 1450 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦) |
27 | 26 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) |
28 | 19, 13, 19, 15, 21, 21, 24, 27 | lemul12ad 12207 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (1 · 1) ≤ (𝑥 · 𝑦)) |
29 | 18, 28 | eqbrtrrid 5183 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ (𝑥 · 𝑦)) |
30 | 16 | ltpnfd 13160 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) < +∞) |
31 | 10, 11, 17, 29, 30 | elicod 13433 | . . . 4 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
32 | 31 | adantl 481 | . . 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 11308 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
38 | fprodge1.ge | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) | |
39 | 36 | ltpnfd 13160 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < +∞) |
40 | 34, 35, 37, 38, 39 | elicod 13433 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (1[,)+∞)) |
41 | 1le1 11888 | . . . . 5 ⊢ 1 ≤ 1 | |
42 | ltpnf 13159 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
43 | 4, 42 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
44 | elico2 13447 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞))) | |
45 | 4, 2, 44 | mp2an 692 | . . . . 5 ⊢ (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞)) |
46 | 4, 41, 43, 45 | mpbir3an 1340 | . . . 4 ⊢ 1 ∈ (1[,)+∞) |
47 | 46 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (1[,)+∞)) |
48 | 3, 9, 32, 33, 40, 47 | fprodcllemf 15990 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) |
49 | icogelb 13434 | . 2 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
50 | 1, 2, 48, 49 | mp3an12i 1464 | 1 ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 Ⅎwnf 1779 ∈ wcel 2105 ⊆ wss 3962 class class class wbr 5147 (class class class)co 7430 Fincfn 8983 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 [,)cico 13385 ∏cprod 15935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-prod 15936 |
This theorem is referenced by: fprodle 16028 |
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