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Theorem efcj 16035

Description: The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)

**Assertion**
Ref	Expression
efcj	⊢ (𝐴 ∈ ℂ → (exp‘(∗‘𝐴)) = (∗‘(exp‘𝐴)))

Proof of Theorem efcj Dummy variables 𝑗 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cjcl 15052	. . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
2		eqid 2733	. . . 4 ⊢ (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))
3	2	efcvg 16028	. . 3 ⊢ ((∗‘𝐴) ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ⇝ (exp‘(∗‘𝐴)))
4	1, 3	syl 17	. 2 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ⇝ (exp‘(∗‘𝐴)))
5		nn0uz 12864	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
6		eqid 2733	. . . 4 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
7	6	efcvg 16028	. . 3 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ⇝ (exp‘𝐴))
8		seqex 13968	. . . 4 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ∈ V
9	8	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ∈ V)
10		0zd 12570	. . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℤ)
11	6	eftval 16020	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘)))
12	11	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘)))
13		eftcl 16017	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ)
14	12, 13	eqeltrd 2834	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
15	5, 10, 14	serf 13996	. . . 4 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))):ℕ₀⟶ℂ)
16	15	ffvelcdmda 7087	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗) ∈ ℂ)
17		addcl 11192	. . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑘 + 𝑚) ∈ ℂ)
18	17	adantl 483	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) ∧ (𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑘 + 𝑚) ∈ ℂ)
19		simpl 484	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → 𝐴 ∈ ℂ)
20		elfznn0 13594	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ₀)
21	19, 20, 14	syl2an 597	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑗)) → ((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
22		simpr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ ℕ₀)
23	22, 5	eleqtrdi 2844	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ (ℤ_≥‘0))
24		cjadd 15088	. . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (∗‘(𝑘 + 𝑚)) = ((∗‘𝑘) + (∗‘𝑚)))
25	24	adantl 483	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) ∧ (𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (∗‘(𝑘 + 𝑚)) = ((∗‘𝑘) + (∗‘𝑚)))
26		expcl 14045	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
27		faccl 14243	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
28	27	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℕ)
29	28	nncnd 12228	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℂ)
30	28	nnne0d 12262	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ≠ 0)
31	26, 29, 30	cjdivd 15170	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘((𝐴↑𝑘) / (!‘𝑘))) = ((∗‘(𝐴↑𝑘)) / (∗‘(!‘𝑘))))
32		cjexp 15097	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
33	28	nnred 12227	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℝ)
34	33	cjred 15173	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘(!‘𝑘)) = (!‘𝑘))
35	32, 34	oveq12d 7427	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((∗‘(𝐴↑𝑘)) / (∗‘(!‘𝑘))) = (((∗‘𝐴)↑𝑘) / (!‘𝑘)))
36	31, 35	eqtrd 2773	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘((𝐴↑𝑘) / (!‘𝑘))) = (((∗‘𝐴)↑𝑘) / (!‘𝑘)))
37	12	fveq2d 6896	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘)) = (∗‘((𝐴↑𝑘) / (!‘𝑘))))
38	2	eftval 16020	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))‘𝑘) = (((∗‘𝐴)↑𝑘) / (!‘𝑘)))
39	38	adantl 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))‘𝑘) = (((∗‘𝐴)↑𝑘) / (!‘𝑘)))
40	36, 37, 39	3eqtr4d 2783	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (∗‘((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘)) = ((𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))
41	19, 20, 40	syl2an 597	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑗)) → (∗‘((𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘)) = ((𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))
42	18, 21, 23, 25, 41	seqhomo 14015	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → (∗‘(seq0( + , (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)) = (seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛))))‘𝑗))
43	42	eqcomd 2739	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛))))‘𝑗) = (∗‘(seq0( + , (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)))
44	5, 7, 9, 10, 16, 43	climcj 15549	. 2 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ⇝ (∗‘(exp‘𝐴)))
45		climuni 15496	. 2 ⊢ ((seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ⇝ (exp‘(∗‘𝐴)) ∧ seq0( + , (𝑛 ∈ ℕ₀ ↦ (((∗‘𝐴)↑𝑛) / (!‘𝑛)))) ⇝ (∗‘(exp‘𝐴))) → (exp‘(∗‘𝐴)) = (∗‘(exp‘𝐴)))
46	4, 44, 45	syl2anc 585	1 ⊢ (𝐴 ∈ ℂ → (exp‘(∗‘𝐴)) = (∗‘(exp‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 + caddc 11113 / cdiv 11871 ℕcn 12212 ℕ₀cn0 12472 ℤ_≥cuz 12822 ...cfz 13484 seqcseq 13966 ↑cexp 14027 !cfa 14233 ∗ccj 15043 ⇝ cli 15428 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-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: resinval 16078 recosval 16079 logcj 26114 cosargd 26116

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