|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > aaliou3lem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for aaliou3 26394. (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 7440 | . . . 4 ⊢ (𝑐 = 𝐴 → (1...𝑐) = (1...𝐴)) | |
| 2 | 1 | sumeq1d 15737 | . . 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 15725 | . . 3 ⊢ Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ V | |
| 5 | 2, 3, 4 | fvmpt 7015 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) = Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏)) | 
| 6 | fzfid 14015 | . . 3 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin) | |
| 7 | elfznn 13594 | . . . . 5 ⊢ (𝑏 ∈ (1...𝐴) → 𝑏 ∈ ℕ) | |
| 8 | 7 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → 𝑏 ∈ ℕ) | 
| 9 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (!‘𝑎) = (!‘𝑏)) | |
| 10 | 9 | negeqd 11503 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → -(!‘𝑎) = -(!‘𝑏)) | 
| 11 | 10 | oveq2d 7448 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (2↑-(!‘𝑎)) = (2↑-(!‘𝑏))) | 
| 12 | aaliou3lem.c | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) | |
| 13 | ovex 7465 | . . . . . 6 ⊢ (2↑-(!‘𝑏)) ∈ V | |
| 14 | 11, 12, 13 | fvmpt 7015 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) = (2↑-(!‘𝑏))) | 
| 15 | 2rp 13040 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
| 16 | nnnn0 12535 | . . . . . . . . . 10 ⊢ (𝑏 ∈ ℕ → 𝑏 ∈ ℕ0) | |
| 17 | 16 | faccld 14324 | . . . . . . . . 9 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℕ) | 
| 18 | 17 | nnzd 12642 | . . . . . . . 8 ⊢ (𝑏 ∈ ℕ → (!‘𝑏) ∈ ℤ) | 
| 19 | 18 | znegcld 12726 | . . . . . . 7 ⊢ (𝑏 ∈ ℕ → -(!‘𝑏) ∈ ℤ) | 
| 20 | rpexpcl 14122 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝑏) ∈ ℤ) → (2↑-(!‘𝑏)) ∈ ℝ+) | |
| 21 | 15, 19, 20 | sylancr 587 | . . . . . 6 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ+) | 
| 22 | 21 | rpred 13078 | . . . . 5 ⊢ (𝑏 ∈ ℕ → (2↑-(!‘𝑏)) ∈ ℝ) | 
| 23 | 14, 22 | eqeltrd 2840 | . . . 4 ⊢ (𝑏 ∈ ℕ → (𝐹‘𝑏) ∈ ℝ) | 
| 24 | 8, 23 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑏 ∈ (1...𝐴)) → (𝐹‘𝑏) ∈ ℝ) | 
| 25 | 6, 24 | fsumrecl 15771 | . 2 ⊢ (𝐴 ∈ ℕ → Σ𝑏 ∈ (1...𝐴)(𝐹‘𝑏) ∈ ℝ) | 
| 26 | 5, 25 | eqeltrd 2840 | 1 ⊢ (𝐴 ∈ ℕ → (𝐻‘𝐴) ∈ ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 1c1 11157 -cneg 11494 ℕcn 12267 2c2 12322 ℤcz 12615 ℝ+crp 13035 ...cfz 13548 ↑cexp 14103 !cfa 14313 Σcsu 15723 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-fac 14314 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 | 
| This theorem is referenced by: aaliou3lem7 26392 aaliou3lem9 26393 | 
| Copyright terms: Public domain | W3C validator |