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Mirrors > Home > MPE Home > Th. List > mbfi1fseqlem1 | Structured version Visualization version GIF version |
Description: Lemma for mbfi1fseq 24325. (Contributed by Mario Carneiro, 16-Aug-2014.) |
Ref | Expression |
---|---|
mbfi1fseq.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
mbfi1fseq.2 | ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
mbfi1fseq.3 | ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) |
Ref | Expression |
---|---|
mbfi1fseqlem1 | ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfi1fseq.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) | |
2 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
3 | ffvelrn 6826 | . . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (0[,)+∞)) | |
4 | 1, 2, 3 | syl2an 598 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞)) |
5 | elrege0 12832 | . . . . . . . . 9 ⊢ ((𝐹‘𝑦) ∈ (0[,)+∞) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑦))) | |
6 | 4, 5 | sylib 221 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑦))) |
7 | 6 | simpld 498 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ) |
8 | 2nn 11698 | . . . . . . . . . 10 ⊢ 2 ∈ ℕ | |
9 | nnnn0 11892 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0) | |
10 | nnexpcl 13438 | . . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ) | |
11 | 8, 9, 10 | sylancr 590 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ) |
12 | 11 | ad2antrl 727 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ) |
13 | 12 | nnred 11640 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ) |
14 | 7, 13 | remulcld 10660 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ) |
15 | reflcl 13161 | . . . . . 6 ⊢ (((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ) |
17 | 16, 12 | nndivred 11679 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ) |
18 | 12 | nnnn0d 11943 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ0) |
19 | 18 | nn0ge0d 11946 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → 0 ≤ (2↑𝑚)) |
20 | mulge0 11147 | . . . . . . . 8 ⊢ ((((𝐹‘𝑦) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑦)) ∧ ((2↑𝑚) ∈ ℝ ∧ 0 ≤ (2↑𝑚))) → 0 ≤ ((𝐹‘𝑦) · (2↑𝑚))) | |
21 | 6, 13, 19, 20 | syl12anc 835 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → 0 ≤ ((𝐹‘𝑦) · (2↑𝑚))) |
22 | flge0nn0 13185 | . . . . . . 7 ⊢ ((((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ ∧ 0 ≤ ((𝐹‘𝑦) · (2↑𝑚))) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℕ0) | |
23 | 14, 21, 22 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℕ0) |
24 | 23 | nn0ge0d 11946 | . . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → 0 ≤ (⌊‘((𝐹‘𝑦) · (2↑𝑚)))) |
25 | 12 | nngt0d 11674 | . . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → 0 < (2↑𝑚)) |
26 | divge0 11498 | . . . . 5 ⊢ ((((⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ ∧ 0 ≤ (⌊‘((𝐹‘𝑦) · (2↑𝑚)))) ∧ ((2↑𝑚) ∈ ℝ ∧ 0 < (2↑𝑚))) → 0 ≤ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) | |
27 | 16, 24, 13, 25, 26 | syl22anc 837 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → 0 ≤ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) |
28 | elrege0 12832 | . . . 4 ⊢ (((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ (0[,)+∞) ↔ (((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ∧ 0 ≤ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))) | |
29 | 17, 27, 28 | sylanbrc 586 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ (0[,)+∞)) |
30 | 29 | ralrimivva 3156 | . 2 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ (0[,)+∞)) |
31 | mbfi1fseq.3 | . . 3 ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) | |
32 | 31 | fmpo 7748 | . 2 ⊢ (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ (0[,)+∞) ↔ 𝐽:(ℕ × ℝ)⟶(0[,)+∞)) |
33 | 30, 32 | sylib 221 | 1 ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℝcr 10525 0cc0 10526 · cmul 10531 +∞cpnf 10661 < clt 10664 ≤ cle 10665 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ0cn0 11885 [,)cico 12728 ⌊cfl 13155 ↑cexp 13425 MblFncmbf 24218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-ico 12732 df-fl 13157 df-seq 13365 df-exp 13426 |
This theorem is referenced by: mbfi1fseqlem5 24323 |
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