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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 11320 | . 2 ⊢ 1 ∈ ℝ* | |
| 2 | pnfxr 11315 | . 2 ⊢ +∞ ∈ ℝ* | |
| 3 | fprodge1.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
| 4 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 5 | icossre 13468 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1[,)+∞) ⊆ ℝ) | |
| 6 | 4, 2, 5 | mp2an 692 | . . . . 5 ⊢ (1[,)+∞) ⊆ ℝ |
| 7 | ax-resscn 11212 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 8 | 6, 7 | sstri 3993 | . . . 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 3979 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ) |
| 13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ) |
| 14 | 6 | sseli 3979 | . . . . . . . 8 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ) |
| 15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
| 16 | 13, 15 | remulcld 11291 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ) |
| 17 | 16 | rexrd 11311 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) ∈ ℝ*) |
| 18 | 1t1e1 12428 | . . . . . 6 ⊢ (1 · 1) = 1 | |
| 19 | 4 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ) |
| 20 | 0le1 11786 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ 1) |
| 22 | icogelb 13438 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥) | |
| 23 | 1, 2, 22 | mp3an12 1453 | . . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥) |
| 24 | 23 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑥) |
| 25 | icogelb 13438 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦) | |
| 26 | 1, 2, 25 | mp3an12 1453 | . . . . . . . 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 12210 | . . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (1 · 1) ≤ (𝑥 · 𝑦)) |
| 29 | 18, 28 | eqbrtrrid 5179 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ (𝑥 · 𝑦)) |
| 30 | 16 | ltpnfd 13163 | . . . . 5 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑦 ∈ (1[,)+∞)) → (𝑥 · 𝑦) < +∞) |
| 31 | 10, 11, 17, 29, 30 | elicod 13437 | . . . 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 11311 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
| 38 | fprodge1.ge | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 1 ≤ 𝐵) | |
| 39 | 36 | ltpnfd 13163 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 < +∞) |
| 40 | 34, 35, 37, 38, 39 | elicod 13437 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (1[,)+∞)) |
| 41 | 1le1 11891 | . . . . 5 ⊢ 1 ≤ 1 | |
| 42 | ltpnf 13162 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
| 43 | 4, 42 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
| 44 | elico2 13451 | . . . . . 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 1342 | . . . 4 ⊢ 1 ∈ (1[,)+∞) |
| 47 | 46 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (1[,)+∞)) |
| 48 | 3, 9, 32, 33, 40, 47 | fprodcllemf 15994 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) |
| 49 | icogelb 13438 | . 2 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (1[,)+∞)) → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
| 50 | 1, 2, 48, 49 | mp3an12i 1467 | 1 ⊢ (𝜑 → 1 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 Ⅎwnf 1783 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 ∏cprod 15939 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 |
| This theorem is referenced by: fprodle 16032 |
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