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Mirrors > Home > MPE Home > Th. List > rpnnen2lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16169. (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 12479 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
2 | 0re 11216 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
3 | 1re 11214 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
4 | 3nn 12291 | . . . . . . . 8 ⊢ 3 ∈ ℕ | |
5 | nndivre 12253 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
6 | 3, 4, 5 | mp2an 691 | . . . . . . 7 ⊢ (1 / 3) ∈ ℝ |
7 | 3re 12292 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
8 | 3pos 12317 | . . . . . . . 8 ⊢ 0 < 3 | |
9 | 7, 8 | recgt0ii 12120 | . . . . . . 7 ⊢ 0 < (1 / 3) |
10 | 2, 6, 9 | ltleii 11337 | . . . . . 6 ⊢ 0 ≤ (1 / 3) |
11 | expge0 14064 | . . . . . . 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 12313 | . . . 4 ⊢ 0 ≤ 0 | |
16 | breq2 5153 | . . . . 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 5153 | . . . . 5 ⊢ (0 = if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑘 ∈ 𝐴, ((1 / 3)↑𝑘), 0))) | |
18 | 16, 17 | ifboth 4568 | . . . 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 3991 | . . . 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 16157 | . . . 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 5177 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝐴)‘𝑘)) |
25 | reexpcl 14044 | . . . . . 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 11217 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 0 ∈ ℝ) | |
29 | simp1 1137 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → 𝐴 ⊆ 𝐵) | |
30 | 29 | sseld 3982 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ 𝐴 → 𝑘 ∈ 𝐵)) |
31 | ifle 13176 | . . . 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 16157 | . . . 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 5181 | . 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 3949 ifcif 4529 𝒫 cpw 4603 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 ≤ cle 11249 / cdiv 11871 ℕcn 12212 3c3 12268 ℕ0cn0 12472 ↑cexp 14027 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-exp 14028 |
This theorem is referenced by: rpnnen2lem5 16161 rpnnen2lem7 16163 rpnnen2lem12 16168 |
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