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Mirrors > Home > MPE Home > Th. List > aaliou3lem5 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 25241. (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 |
---|---|
aaliou3lem5 | ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7218 | . . . 4 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴)) | |
2 | 1 | sumeq1d 15262 | . . 3 ⊢ (𝑐 = 𝐴 → Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏)) |
3 | aaliou3lem.e | . . 3 ⊢ 𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹‘𝑏)) | |
4 | sumex 15248 | . . 3 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V | |
5 | 2, 3, 4 | fvmpt 6815 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏)) |
6 | fzfid 13543 | . . 3 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin) | |
7 | elfznn 13138 | . . . . 5 ⊢ (𝑏 ∈ (1...𝐴) → 𝑏 ∈ ℕ) | |
8 | 7 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ∈ ℕ) |
9 | fveq2 6714 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏)) | |
10 | 9 | negeqd 11069 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏)) |
11 | 10 | oveq2d 7226 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏))) |
12 | aaliou3lem.c | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) | |
13 | ovex 7243 | . . . . . 6 ⊢ (2↑-(!‘𝑏)) ∈ V | |
14 | 11, 12, 13 | fvmpt 6815 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏))) |
15 | 2rp 12588 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
16 | nnnn0 12094 | . . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ0) | |
17 | 16 | faccld 13847 | . . . . . . . . 9 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ) |
18 | 17 | nnzd 12278 | . . . . . . . 8 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ) |
19 | 18 | znegcld 12281 | . . . . . . 7 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ) |
20 | rpexpcl 13651 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ+) | |
21 | 15, 19, 20 | sylancr 590 | . . . . . 6 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ+) |
22 | 21 | rpred 12625 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ) |
23 | 14, 22 | eqeltrd 2838 | . . . 4 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℝ) |
24 | 8, 23 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (𝐹‘𝑏) ∈ ℝ) |
25 | 6, 24 | fsumrecl 15295 | . 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ ℝ) |
26 | 5, 25 | eqeltrd 2838 | 1 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5132 ‘cfv 6377 (class class class)co 7210 ℝcr 10725 1c1 10727 -cneg 11060 ℕcn 11827 2c2 11882 ℤcz 12173 ℝ+crp 12583 ...cfz 13092 ↑cexp 13632 !cfa 13836 Σcsu 15246 |
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 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-inf2 9253 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-se 5507 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-isom 6386 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-sup 9055 df-oi 9123 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fz 13093 df-fzo 13236 df-seq 13572 df-exp 13633 df-fac 13837 df-hash 13894 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-clim 15046 df-sum 15247 |
This theorem is referenced by: aaliou3lem7 25239 aaliou3lem9 25240 |
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