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Mirrors > Home > MPE Home > Th. List > aaliou3lem5 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 26299. (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 7421 | . . . 4 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴)) | |
2 | 1 | sumeq1d 15674 | . . 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 15661 | . . 3 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V | |
5 | 2, 3, 4 | fvmpt 6998 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏)) |
6 | fzfid 13965 | . . 3 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin) | |
7 | elfznn 13557 | . . . . 5 ⊢ (𝑏 ∈ (1...𝐴) → 𝑏 ∈ ℕ) | |
8 | 7 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ∈ ℕ) |
9 | fveq2 6890 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏)) | |
10 | 9 | negeqd 11479 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏)) |
11 | 10 | oveq2d 7429 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏))) |
12 | aaliou3lem.c | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) | |
13 | ovex 7446 | . . . . . 6 ⊢ (2↑-(!‘𝑏)) ∈ V | |
14 | 11, 12, 13 | fvmpt 6998 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏))) |
15 | 2rp 13006 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
16 | nnnn0 12504 | . . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ0) | |
17 | 16 | faccld 14270 | . . . . . . . . 9 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ) |
18 | 17 | nnzd 12610 | . . . . . . . 8 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ) |
19 | 18 | znegcld 12693 | . . . . . . 7 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ) |
20 | rpexpcl 14072 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ+) | |
21 | 15, 19, 20 | sylancr 585 | . . . . . 6 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ+) |
22 | 21 | rpred 13043 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ) |
23 | 14, 22 | eqeltrd 2825 | . . . 4 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℝ) |
24 | 8, 23 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (𝐹‘𝑏) ∈ ℝ) |
25 | 6, 24 | fsumrecl 15707 | . 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ ℝ) |
26 | 5, 25 | eqeltrd 2825 | 1 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5227 ‘cfv 6543 (class class class)co 7413 ℝcr 11132 1c1 11134 -cneg 11470 ℕcn 12237 2c2 12292 ℤcz 12583 ℝ+crp 13001 ...cfz 13511 ↑cexp 14053 !cfa 14259 Σcsu 15659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-fac 14260 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 |
This theorem is referenced by: aaliou3lem7 26297 aaliou3lem9 26298 |
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