efadd - Metamath Proof Explorer

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Theorem efadd 15655

Description: Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

**Assertion**
Ref	Expression
efadd	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))

Proof of Theorem efadd Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
2		eqid 2737	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝐵↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝐵↑𝑛) / (!‘𝑛)))
3		eqid 2737	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛)))
4		simpl 486	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
5		simpr 488	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
6	1, 2, 3, 4, 5	efaddlem 15654	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 + caddc 10732 · cmul 10734 / cdiv 11489 ℕ₀cn0 12090 ↑cexp 13635 !cfa 13839 expce 15623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-ico 12941 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629

This theorem is referenced by: fprodefsum 15656 efcan 15657 efsub 15661 efexp 15662 eflt 15678 efeul 15723 sinadd 15725 cosadd 15726 absef 15758 efieq1re 15760 dvef 24877 reefgim 25342 efper 25369 sineq0 25413 efgh 25430 efif1olem4 25434 eff1olem 25437 logneg 25476 lognegb 25478 relogmul 25480 eflogeq 25490 logimul 25502 logmul2 25504 efopn 25546 cxpadd 25567 mulcxp 25573 cxpsqrt 25591 abscxpbnd 25639 cxpeq 25643 ang180lem1 25692 efiatan2 25800 gamcvg 25938 gamp1 25940 gamcvg2lem 25941 efnnfsumcl 25985 efchtdvds 26041 prmorcht 26060 chtublem 26092 bposlem9 26173 pntibndlem3 26473 circlemeth 32332 iprodefisumlem 33424 sineq0ALT 42230

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