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Mirrors > Home > ILE Home > Th. List > fprodge1 | 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 7937 | . 2 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 7931 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge1.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | 1re 7878 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | icossre 9859 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1[,)+∞) ⊆ ℝ) | |
6 | 4, 2, 5 | mp2an 423 | . . . . 5 ⊢ (1[,)+∞) ⊆ ℝ |
7 | ax-resscn 7825 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
8 | 6, 7 | sstri 3137 | . . . 4 ⊢ (1[,)+∞) ⊆ ℂ |
9 | 8 | a1i 9 | . . 3 ⊢ (𝜑 → (1[,)+∞) ⊆ ℂ) |
10 | 1 | a1i 9 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ*) |
11 | 2 | a1i 9 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → +∞ ∈ ℝ*) |
12 | 6 | sseli 3124 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ) |
13 | 12 | adantr 274 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ) |
14 | 6 | sseli 3124 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ) |
15 | 14 | adantl 275 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
16 | 13, 15 | remulcld 7909 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ) |
17 | 16 | rexrd 7928 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ*) |
18 | 1t1e1 8986 | . . . . . 6 ⊢ (1 · 1) = 1 | |
19 | 4 | a1i 9 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ) |
20 | 0le1 8357 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
21 | 20 | a1i 9 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ 1) |
22 | icogelb 10169 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥) | |
23 | 1, 2, 22 | mp3an12 1309 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥) |
24 | 23 | adantr 274 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑥) |
25 | icogelb 10169 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) | |
26 | 1, 2, 25 | mp3an12 1309 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦) |
27 | 26 | adantl 275 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) |
28 | 19, 13, 19, 15, 21, 21, 24, 27 | lemul12ad 8814 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (1 · 1) ≤ (𝑥 · 𝑦)) |
29 | 18, 28 | eqbrtrrid 4001 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ (𝑥 · 𝑦)) |
30 | 16 | ltpnfd 9689 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) < +∞) |
31 | 10, 11, 17, 29, 30 | elicod 10168 | . . . 4 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
32 | 31 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞))) → (𝑥 · 𝑦) ∈ (1[,)+∞)) |
33 | fprodge1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
34 | 1 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℝ*) |
35 | 2 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → +∞ ∈ ℝ*) |
36 | fprodge1.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
37 | 36 | rexrd 7928 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
38 | fprodge1.ge | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) | |
39 | 36 | ltpnfd 9689 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < +∞) |
40 | 34, 35, 37, 38, 39 | elicod 10168 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (1[,)+∞)) |
41 | 1le1 8448 | . . . . 5 ⊢ 1 ≤ 1 | |
42 | ltpnf 9688 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
43 | 4, 42 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
44 | elico2 9842 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞))) | |
45 | 4, 2, 44 | mp2an 423 | . . . . 5 ⊢ (1 ∈ (1[,)+∞) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 < +∞)) |
46 | 4, 41, 43, 45 | mpbir3an 1164 | . . . 4 ⊢ 1 ∈ (1[,)+∞) |
47 | 46 | a1i 9 | . . 3 ⊢ (𝜑 → 1 ∈ (1[,)+∞)) |
48 | 3, 9, 32, 33, 40, 47 | fprodcllemf 11514 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) |
49 | icogelb 10169 | . 2 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
50 | 1, 2, 48, 49 | mp3an12i 1323 | 1 ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 Ⅎwnf 1440 ∈ wcel 2128 ⊆ wss 3102 class class class wbr 3966 (class class class)co 5825 Fincfn 6686 ℂcc 7731 ℝcr 7732 0cc0 7733 1c1 7734 · cmul 7738 +∞cpnf 7910 ℝ*cxr 7912 < clt 7913 ≤ cle 7914 [,)cico 9795 ∏cprod 11451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-ico 9799 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-ihash 10654 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-proddc 11452 |
This theorem is referenced by: (None) |
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