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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > expfac | Structured version Visualization version GIF version |
Description: Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
expfac.f | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
expfac | ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12759 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12469 | . 2 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℤ) | |
3 | expfac.f | . . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | nn0ex 12377 | . . . . 5 ⊢ ℕ0 ∈ V | |
5 | 4 | mptex 7169 | . . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) ∈ V |
6 | 3, 5 | eqeltri 2834 | . . 3 ⊢ 𝐹 ∈ V |
7 | 6 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ V) |
8 | 3 | efcllem 15920 | . 2 ⊢ (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ ) |
9 | oveq2 7359 | . . . . 5 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚)) | |
10 | fveq2 6839 | . . . . 5 ⊢ (𝑛 = 𝑚 → (!‘𝑛) = (!‘𝑚)) | |
11 | 9, 10 | oveq12d 7369 | . . . 4 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑚) / (!‘𝑚))) |
12 | simpr 485 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0) | |
13 | eftcl 15916 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → ((𝐴↑𝑚) / (!‘𝑚)) ∈ ℂ) | |
14 | 3, 11, 12, 13 | fvmptd3 6968 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) = ((𝐴↑𝑚) / (!‘𝑚))) |
15 | 14, 13 | eqeltrd 2838 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ) |
16 | 1, 2, 7, 8, 15 | serf0 15525 | 1 ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 0cc0 11009 / cdiv 11770 ℕ0cn0 12371 ↑cexp 13921 !cfa 14127 ⇝ cli 15326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-ico 13224 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-fac 14128 df-hash 14185 df-shft 14912 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-limsup 15313 df-clim 15330 df-rlim 15331 df-sum 15531 |
This theorem is referenced by: etransclem48 44424 |
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