iprodefisum - Mathbox for Scott Fenton

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Theorem iprodefisum 33222

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 15164	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
7		efne0 15498	. . 3 ⊢ (Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
9		efcn 25137	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
10	9	a1i 11	. . . 4 ⊢ (𝜑 → exp ∈ (ℂ–cn→ℂ))
11		fveq2 6658	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
12		eqid 2758	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))
13		fvex 6671	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
14	11, 12, 13	fvmpt 6759	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
15	14	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
16	3, 4	eqeltrd 2852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
17	15, 16	eqeltrd 2852	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) ∈ ℂ)
18	1, 2, 17	serf 13448	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))):𝑍⟶ℂ)
19	1	eqcomi 2767	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) = 𝑍
20	14, 19	eleq2s 2870	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
21	20	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) = (𝐹‘𝑘))
22	2, 21	seqfeq 13445	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) = seq𝑀( + , 𝐹))
23		climdm 14959	. . . . . 6 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
24	5, 23	sylib 221	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
25	22, 24	eqbrtrd 5054	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
26		climcl 14904	. . . . 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 23601	. . 3 ⊢ (𝜑 → (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
29	11	cbvmptv 5135	. . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))
30	16, 29	fmptd 6869	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)):𝑍⟶ℂ)
31	1, 2, 30	iprodefisumlem 33221	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) = (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))))
32	1, 2, 3, 4	isum 15124	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
33	32	fveq2d 6662	. . 3 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) = (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
34	28, 31, 33	3brtr4d 5064	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘Σ𝑘 ∈ 𝑍 𝐵))
35		fvco3 6751	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)))
37	15	fveq2d 6662	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)) = (exp‘(𝐹‘𝑘)))
38	3	fveq2d 6662	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) = (exp‘𝐵))
39	36, 37, 38	3eqtrd 2797	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘𝐵))
40		efcl 15484	. . 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 15342	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 class class class wbr 5032 ↦ cmpt 5112 dom cdm 5524 ∘ ccom 5528 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 0cc0 10575 + caddc 10578 · cmul 10580 ℤcz 12020 ℤ_≥cuz 12282 seqcseq 13418 ⇝ cli 14889 Σcsu 15090 ∏cprod 15307 expce 15463 –cn→ccncf 23577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-seq 13419 df-exp 13480 df-fac 13684 df-bc 13713 df-hash 13741 df-shft 14474 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-limsup 14876 df-clim 14893 df-rlim 14894 df-sum 15091 df-prod 15308 df-ef 15469 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-limc 24565 df-dv 24566

This theorem is referenced by: iprodgam 33223

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