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Mirrors > Home > MPE Home > Th. List > aaliou3lem4 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 25268. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Ref | Expression |
---|---|
aaliou3lem.c | ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) |
aaliou3lem.d | ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) |
aaliou3lem.e | ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) |
Ref | Expression |
---|---|
aaliou3lem4 | ⊢ 𝐿 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aaliou3lem.d | . . 3 ⊢ 𝐿 = Σ𝑏 ∈ ℕ (𝐹‘𝑏) | |
2 | nnuz 12501 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
3 | 2 | sumeq1i 15286 | . . 3 ⊢ Σ𝑏 ∈ ℕ (𝐹‘𝑏) = Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) |
4 | 1, 3 | eqtri 2766 | . 2 ⊢ 𝐿 = Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) |
5 | 1nn 11865 | . . 3 ⊢ 1 ∈ ℕ | |
6 | eqid 2738 | . . . . 5 ⊢ (𝑐 ∈ (ℤ≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) = (𝑐 ∈ (ℤ≥‘1) ↦ ((2↑-(!‘1)) · ((1 / 2)↑(𝑐 − 1)))) | |
7 | aaliou3lem.c | . . . . 5 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) | |
8 | 6, 7 | aaliou3lem3 25261 | . . . 4 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1))))) |
9 | 8 | simp2d 1145 | . . 3 ⊢ (1 ∈ ℕ → Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ+) |
10 | rpre 12618 | . . 3 ⊢ (Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ+ → Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ) | |
11 | 5, 9, 10 | mp2b 10 | . 2 ⊢ Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ |
12 | 4, 11 | eqeltri 2835 | 1 ⊢ 𝐿 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2111 class class class wbr 5067 ↦ cmpt 5149 dom cdm 5565 ‘cfv 6397 (class class class)co 7231 ℝcr 10752 1c1 10754 + caddc 10756 · cmul 10758 ≤ cle 10892 − cmin 11086 -cneg 11087 / cdiv 11513 ℕcn 11854 2c2 11909 ℤ≥cuz 12462 ℝ+crp 12610 ...cfz 13119 seqcseq 13598 ↑cexp 13659 !cfa 13863 ⇝ cli 15069 Σcsu 15273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-pm 8531 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-sup 9082 df-inf 9083 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-n0 12115 df-z 12201 df-uz 12463 df-rp 12611 df-ioc 12964 df-ico 12965 df-fz 13120 df-fzo 13263 df-fl 13391 df-seq 13599 df-exp 13660 df-fac 13864 df-hash 13921 df-shft 14654 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-limsup 15056 df-clim 15073 df-rlim 15074 df-sum 15274 |
This theorem is referenced by: aaliou3lem7 25266 aaliou3lem9 25267 |
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