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Mirrors > Home > MPE Home > Th. List > rpnnen2lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16115. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.) |
Ref | Expression |
---|---|
rpnnen2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) |
Ref | Expression |
---|---|
rpnnen2lem4 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘𝐵)‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 12427 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | 0re 11164 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
3 | 1re 11162 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
4 | 3nn 12239 | . . . . . . . 8 ⊢ 3 ∈ ℕ | |
5 | nndivre 12201 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 691 | . . . . . . 7 ⊢ (1 / 3) ∈ ℝ |
7 | 3re 12240 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
8 | 3pos 12265 | . . . . . . . 8 ⊢ 0 < 3 | |
9 | 7, 8 | recgt0ii 12068 | . . . . . . 7 ⊢ 0 < (1 / 3) |
10 | 2, 6, 9 | ltleii 11285 | . . . . . 6 ⊢ 0 ≤ (1 / 3) |
11 | expge0 14011 | . . . . . . 7 ⊢ (((1 / 3) ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ (1 / 3)) → 0 ≤ ((1 / 3)↑𝑘)) | |
12 | 6, 11 | mp3an1 1449 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ 0 ≤ (1 / 3)) → 0 ≤ ((1 / 3)↑𝑘)) |
13 | 1, 10, 12 | sylancl 587 | . . . . 5 ⊢ (𝑘 ∈ ℕ → 0 ≤ ((1 / 3)↑𝑘)) |
14 | 13 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((1 / 3)↑𝑘)) |
15 | 0le0 12261 | . . . 4 ⊢ 0 ≤ 0 | |
16 | breq2 5114 | . . . . 5 ⊢ (((1 / 3)↑𝑘) = if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0) → (0 ≤ ((1 / 3)↑𝑘) ↔ 0 ≤ if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0))) | |
17 | breq2 5114 | . . . . 5 ⊢ (0 = if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0))) | |
18 | 16, 17 | ifboth 4530 | . . . 4 ⊢ ((0 ≤ ((1 / 3)↑𝑘) ∧ 0 ≤ 0) → 0 ≤ if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0)) |
19 | 14, 15, 18 | sylancl 587 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0)) |
20 | sstr 3957 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ) → 𝐴 ⊆ ℕ) | |
21 | rpnnen2.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) | |
22 | 21 | rpnnen2lem1 16103 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) = if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0)) |
23 | 20, 22 | stoic3 1779 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) = if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0)) |
24 | 19, 23 | breqtrrd 5138 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝐴)‘𝑘)) |
25 | reexpcl 13991 | . . . . . 6 ⊢ (((1 / 3) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((1 / 3)↑𝑘) ∈ ℝ) | |
26 | 6, 1, 25 | sylancr 588 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / 3)↑𝑘) ∈ ℝ) |
27 | 26 | 3ad2ant3 1136 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((1 / 3)↑𝑘) ∈ ℝ) |
28 | 0red 11165 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ∈ ℝ) | |
29 | simp1 1137 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 𝐴 ⊆ 𝐵) | |
30 | 29 | sseld 3948 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ 𝐴 → 𝑘 ∈ 𝐵)) |
31 | ifle 13123 | . . . 4 ⊢ (((((1 / 3)↑𝑘) ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ≤ ((1 / 3)↑𝑘)) ∧ (𝑘 ∈ 𝐴 → 𝑘 ∈ 𝐵)) → if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0) ≤ if(𝑘 ∈ 𝐵, ((1 / 3)↑𝑘), 0)) | |
32 | 27, 28, 14, 30, 31 | syl31anc 1374 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0) ≤ if(𝑘 ∈ 𝐵, ((1 / 3)↑𝑘), 0)) |
33 | 21 | rpnnen2lem1 16103 | . . . 4 ⊢ ((𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐵)‘𝑘) = if(𝑘 ∈ 𝐵, ((1 / 3)↑𝑘), 0)) |
34 | 33 | 3adant1 1131 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐵)‘𝑘) = if(𝑘 ∈ 𝐵, ((1 / 3)↑𝑘), 0)) |
35 | 32, 23, 34 | 3brtr4d 5142 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘𝐵)‘𝑘)) |
36 | 24, 35 | jca 513 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘𝐵)‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 ifcif 4491 𝒫 cpw 4565 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 ≤ cle 11197 / cdiv 11819 ℕcn 12160 3c3 12216 ℕ0cn0 12420 ↑cexp 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-seq 13914 df-exp 13975 |
This theorem is referenced by: rpnnen2lem5 16107 rpnnen2lem7 16109 rpnnen2lem12 16114 |
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