efadd - Metamath Proof Explorer

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Theorem efadd 16037

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 2733	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
2		eqid 2733	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝐵↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝐵↑𝑛) / (!‘𝑛)))
3		eqid 2733	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (((𝐴 + 𝐵)↑𝑛) / (!‘𝑛)))
4		simpl 484	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
5		simpr 486	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
6	1, 2, 3, 4, 5	efaddlem 16036	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + 𝐵)) = ((exp‘𝐴) · (exp‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 + caddc 11113 · cmul 11115 / cdiv 11871 ℕ₀cn0 12472 ↑cexp 14027 !cfa 14233 expce 16005

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011

This theorem is referenced by: fprodefsum 16038 efcan 16039 efsub 16043 efexp 16044 eflt 16060 efeul 16105 sinadd 16107 cosadd 16108 absef 16140 efieq1re 16142 dvef 25497 reefgim 25962 efper 25989 sineq0 26033 efgh 26050 efif1olem4 26054 eff1olem 26057 logneg 26096 lognegb 26098 relogmul 26100 eflogeq 26110 logimul 26122 logmul2 26124 efopn 26166 cxpadd 26187 mulcxp 26193 cxpsqrt 26211 abscxpbnd 26261 cxpeq 26265 ang180lem1 26314 efiatan2 26422 gamcvg 26560 gamp1 26562 gamcvg2lem 26563 efnnfsumcl 26607 efchtdvds 26663 prmorcht 26682 chtublem 26714 bposlem9 26795 pntibndlem3 27095 circlemeth 33683 iprodefisumlem 34741 sineq0ALT 43746

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