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Mirrors > Home > ILE Home > Th. List > eft0val | GIF version |
Description: The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
eft0val | ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exp0 10427 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | |
2 | 1 | oveq1d 5840 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = (1 / (!‘0))) |
3 | fac0 10606 | . . . 4 ⊢ (!‘0) = 1 | |
4 | 3 | oveq2i 5836 | . . 3 ⊢ (1 / (!‘0)) = (1 / 1) |
5 | 1div1e1 8578 | . . 3 ⊢ (1 / 1) = 1 | |
6 | 4, 5 | eqtri 2178 | . 2 ⊢ (1 / (!‘0)) = 1 |
7 | 2, 6 | eqtrdi 2206 | 1 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 ‘cfv 5171 (class class class)co 5825 ℂcc 7731 0cc0 7733 1c1 7734 / cdiv 8546 ↑cexp 10422 !cfa 10603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-seqfrec 10349 df-exp 10423 df-fac 10604 |
This theorem is referenced by: ef4p 11595 efgt1p2 11596 efgt1p 11597 dveflem 13129 |
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