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Mirrors > Home > MPE Home > Th. List > rpnnen2lem5 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 14794. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
rpnnen2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) |
Ref | Expression |
---|---|
rpnnen2lem5 | ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → seq𝑀( + , (𝐹‘𝐴)) ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11599 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1nn 10908 | . . . . 5 ⊢ 1 ∈ ℕ | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ ℕ → 1 ∈ ℕ) |
4 | ssid 3587 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
5 | rpnnen2.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) | |
6 | 5 | rpnnen2lem2 14783 | . . . . . 6 ⊢ (ℕ ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ) |
7 | 4, 6 | mp1i 13 | . . . . 5 ⊢ (𝐴 ⊆ ℕ → (𝐹‘ℕ):ℕ⟶ℝ) |
8 | 7 | ffvelrnda 6267 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) ∈ ℝ) |
9 | 5 | rpnnen2lem2 14783 | . . . . 5 ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴):ℕ⟶ℝ) |
10 | 9 | ffvelrnda 6267 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) ∈ ℝ) |
11 | 5 | rpnnen2lem3 14784 | . . . . 5 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
12 | seqex 12665 | . . . . . 6 ⊢ seq1( + , (𝐹‘ℕ)) ∈ V | |
13 | ovex 6577 | . . . . . 6 ⊢ (1 / 2) ∈ V | |
14 | 12, 13 | breldm 5251 | . . . . 5 ⊢ (seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) → seq1( + , (𝐹‘ℕ)) ∈ dom ⇝ ) |
15 | 11, 14 | mp1i 13 | . . . 4 ⊢ (𝐴 ⊆ ℕ → seq1( + , (𝐹‘ℕ)) ∈ dom ⇝ ) |
16 | elnnuz 11600 | . . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1)) | |
17 | 5 | rpnnen2lem4 14785 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℕ ∧ ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘))) |
18 | 4, 17 | mp3an2 1404 | . . . . . 6 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘))) |
19 | 16, 18 | sylan2br 492 | . . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (0 ≤ ((𝐹‘𝐴)‘𝑘) ∧ ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘))) |
20 | 19 | simpld 474 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → 0 ≤ ((𝐹‘𝐴)‘𝑘)) |
21 | 19 | simprd 478 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘𝐴)‘𝑘) ≤ ((𝐹‘ℕ)‘𝑘)) |
22 | 1, 3, 8, 10, 15, 20, 21 | cvgcmp 14389 | . . 3 ⊢ (𝐴 ⊆ ℕ → seq1( + , (𝐹‘𝐴)) ∈ dom ⇝ ) |
23 | 22 | adantr 480 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → seq1( + , (𝐹‘𝐴)) ∈ dom ⇝ ) |
24 | simpr 476 | . . 3 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ) | |
25 | 10 | adantlr 747 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) ∈ ℝ) |
26 | 25 | recnd 9947 | . . 3 ⊢ (((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝐴)‘𝑘) ∈ ℂ) |
27 | 1, 24, 26 | iserex 14235 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → (seq1( + , (𝐹‘𝐴)) ∈ dom ⇝ ↔ seq𝑀( + , (𝐹‘𝐴)) ∈ dom ⇝ )) |
28 | 23, 27 | mpbid 221 | 1 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑀 ∈ ℕ) → seq𝑀( + , (𝐹‘𝐴)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 ℤ≥cuz 11563 seqcseq 12663 ↑cexp 12722 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 |
This theorem is referenced by: rpnnen2lem6 14787 rpnnen2lem7 14788 rpnnen2lem8 14789 rpnnen2lem9 14790 rpnnen2lem12 14793 |
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