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Mirrors > Home > MPE Home > Th. List > efgt1p | Structured version Visualization version GIF version |
Description: The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
efgt1p | ⊢ (𝐴 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcn 12930 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
2 | nn0uz 12810 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0nn0 12433 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℕ0) |
5 | 1e0p1 12665 | . . . 4 ⊢ 1 = (0 + 1) | |
6 | 0z 12515 | . . . . 5 ⊢ 0 ∈ ℤ | |
7 | eqid 2733 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
8 | 7 | eftval 15964 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0))) |
9 | 3, 8 | ax-mp 5 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0)) |
10 | eft0val 15999 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
11 | 9, 10 | eqtrid 2785 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
12 | 6, 11 | seq1i 13926 | . . . 4 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
13 | 1nn0 12434 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
14 | 7 | eftval 15964 | . . . . . 6 ⊢ (1 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1))) |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1)) |
16 | fac1 14183 | . . . . . . 7 ⊢ (!‘1) = 1 | |
17 | 16 | oveq2i 7369 | . . . . . 6 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
18 | exp1 13979 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
19 | 18 | oveq1d 7373 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
20 | div1 11849 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
21 | 19, 20 | eqtrd 2773 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
22 | 17, 21 | eqtrid 2785 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
23 | 15, 22 | eqtrid 2785 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
24 | 2, 4, 5, 12, 23 | seqp1d 13929 | . . 3 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
25 | 1, 24 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
26 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
27 | 13 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℕ0) |
28 | 7, 26, 27 | effsumlt 15998 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) < (exp‘𝐴)) |
29 | 25, 28 | eqbrtrrd 5130 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 ℂcc 11054 0cc0 11056 1c1 11057 + caddc 11059 < clt 11194 / cdiv 11817 ℕ0cn0 12418 ℝ+crp 12920 seqcseq 13912 ↑cexp 13973 !cfa 14179 expce 15949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-fac 14180 df-hash 14237 df-shft 14958 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-ef 15955 |
This theorem is referenced by: efgt1 16003 reeff1olem 25821 logdivlti 25991 logdifbnd 26359 emcllem4 26364 harmonicbnd4 26376 |
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