iprodefisum - Mathbox for Scott Fenton

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Theorem iprodefisum 35728

Description: Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)

**Hypotheses**
Ref	Expression
iprodefisum.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
iprodefisum.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
iprodefisum.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
iprodefisum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
iprodefisum.5	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

**Assertion**
Ref	Expression
iprodefisum	⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))

Distinct variable groups: 𝑘,𝐹 𝜑,𝑘 𝑘,𝑀 𝑘,𝑍 Allowed substitution hint: 𝐵(𝑘)

Proof of Theorem iprodefisum Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodefisum.1	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		iprodefisum.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		iprodefisum.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)
4		iprodefisum.4	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
5		iprodefisum.5	. . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
6	1, 2, 3, 4, 5	isumcl 15727	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
7		efne0 16064	. . 3 ⊢ (Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
9		efcn 26353	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
10	9	a1i 11	. . . 4 ⊢ (𝜑 → exp ∈ (ℂ–cn→ℂ))
11		fveq2 6858	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
12		eqid 2729	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))
13		fvex 6871	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
14	11, 12, 13	fvmpt 6968	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
15	14	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
16	3, 4	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
17	15, 16	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) ∈ ℂ)
18	1, 2, 17	serf 13995	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))):𝑍⟶ℂ)
19	1	eqcomi 2738	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) = 𝑍
20	14, 19	eleq2s 2846	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
21	20	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
22	2, 21	seqfeq 13992	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) = seq𝑀( + , 𝐹))
23		climdm 15520	. . . . . 6 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
24	5, 23	sylib 218	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
25	22, 24	eqbrtrd 5129	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
26		climcl 15465	. . . . 5 ⊢ (seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)) → ( ⇝ ‘seq𝑀( + , 𝐹)) ∈ ℂ)
27	24, 26	syl 17	. . . 4 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , 𝐹)) ∈ ℂ)
28	1, 2, 10, 18, 25, 27	climcncf 24793	. . 3 ⊢ (𝜑 → (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
29	11	cbvmptv 5211	. . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))
30	16, 29	fmptd 7086	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)):𝑍⟶ℂ)
31	1, 2, 30	iprodefisumlem 35727	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) = (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))))
32	1, 2, 3, 4	isum 15685	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
33	32	fveq2d 6862	. . 3 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) = (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
34	28, 31, 33	3brtr4d 5139	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘Σ𝑘 ∈ 𝑍 𝐵))
35		fvco3 6960	. . . 4 ⊢ (((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)):𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)))
36	30, 35	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)))
37	15	fveq2d 6862	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)) = (exp‘(𝐹‘𝑘)))
38	3	fveq2d 6862	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) = (exp‘𝐵))
39	36, 37, 38	3eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘𝐵))
40		efcl 16048	. . 3 ⊢ (𝐵 ∈ ℂ → (exp‘𝐵) ∈ ℂ)
41	4, 40	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘𝐵) ∈ ℂ)
42	1, 2, 8, 34, 39, 41	iprodn0 15906	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 0cc0 11068 + caddc 11071 · cmul 11073 ℤcz 12529 ℤ_≥cuz 12793 seqcseq 13966 ⇝ cli 15450 Σcsu 15652 ∏cprod 15869 expce 16027 –cn→ccncf 24769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-prod 15870 df-ef 16033 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768

This theorem is referenced by: iprodgam 35729

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