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Mirrors > Home > MPE Home > Th. List > aaliou3lem4 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 25539. (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 12649 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
3 | 2 | sumeq1i 15438 | . . 3 ⊢ Σ𝑏 ∈ ℕ (𝐹‘𝑏) = Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) |
4 | 1, 3 | eqtri 2761 | . 2 ⊢ 𝐿 = Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) |
5 | 1nn 12012 | . . 3 ⊢ 1 ∈ ℕ | |
6 | eqid 2733 | . . . . 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 25532 | . . . 4 ⊢ (1 ∈ ℕ → (seq1( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘1))))) |
9 | 8 | simp2d 1141 | . . 3 ⊢ (1 ∈ ℕ → Σ𝑏 ∈ (ℤ≥‘1)(𝐹‘𝑏) ∈ ℝ+) |
10 | rpre 12766 | . . 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 2830 | 1 ⊢ 𝐿 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2101 class class class wbr 5077 ↦ cmpt 5160 dom cdm 5591 ‘cfv 6447 (class class class)co 7295 ℝcr 10898 1c1 10900 + caddc 10902 · cmul 10904 ≤ cle 11038 − cmin 11233 -cneg 11234 / cdiv 11660 ℕcn 12001 2c2 12056 ℤ≥cuz 12610 ℝ+crp 12758 ...cfz 13267 seqcseq 13749 ↑cexp 13810 !cfa 14015 ⇝ cli 15221 Σcsu 15425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-pm 8638 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-inf 9230 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-z 12348 df-uz 12611 df-rp 12759 df-ioc 13112 df-ico 13113 df-fz 13268 df-fzo 13411 df-fl 13540 df-seq 13750 df-exp 13811 df-fac 14016 df-hash 14073 df-shft 14806 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-limsup 15208 df-clim 15225 df-rlim 15226 df-sum 15426 |
This theorem is referenced by: aaliou3lem7 25537 aaliou3lem9 25538 |
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