iprodefisum - Mathbox for Scott Fenton

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Theorem iprodefisum 35942

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 15717	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
7		efne0 16057	. . 3 ⊢ (Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
9		efcn 26424	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
10	9	a1i 11	. . . 4 ⊢ (𝜑 → exp ∈ (ℂ–cn→ℂ))
11		fveq2 6835	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
12		eqid 2737	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))
13		fvex 6848	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
14	11, 12, 13	fvmpt 6942	. . . . . . 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 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
17	15, 16	eqeltrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) ∈ ℂ)
18	1, 2, 17	serf 13986	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))):𝑍⟶ℂ)
19	1	eqcomi 2746	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) = 𝑍
20	14, 19	eleq2s 2855	. . . . . . 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 13983	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) = seq𝑀( + , 𝐹))
23		climdm 15510	. . . . . 6 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
24	5, 23	sylib 218	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
25	22, 24	eqbrtrd 5108	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
26		climcl 15455	. . . . 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 24880	. . 3 ⊢ (𝜑 → (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
29	11	cbvmptv 5190	. . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))
30	16, 29	fmptd 7061	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)):𝑍⟶ℂ)
31	1, 2, 30	iprodefisumlem 35941	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) = (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))))
32	1, 2, 3, 4	isum 15675	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
33	32	fveq2d 6839	. . 3 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) = (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
34	28, 31, 33	3brtr4d 5118	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘Σ𝑘 ∈ 𝑍 𝐵))
35		fvco3 6934	. . . 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 581	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)))
37	15	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)) = (exp‘(𝐹‘𝑘)))
38	3	fveq2d 6839	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) = (exp‘𝐵))
39	36, 37, 38	3eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘𝐵))
40		efcl 16041	. . 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 15899	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5625 ∘ ccom 5629 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 0cc0 11032 + caddc 11035 · cmul 11037 ℤcz 12518 ℤ_≥cuz 12782 seqcseq 13957 ⇝ cli 15440 Σcsu 15642 ∏cprod 15862 expce 16020 –cn→ccncf 24856

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-prod 15863 df-ef 16026 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847

This theorem is referenced by: iprodgam 35943

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