iprodefisum - Mathbox for Scott Fenton

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Theorem iprodefisum 35792

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 15674	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ)
7		efne0 16011	. . 3 ⊢ (Σ𝑘 ∈ 𝑍 𝐵 ∈ ℂ → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) ≠ 0)
9		efcn 26386	. . . . 5 ⊢ exp ∈ (ℂ–cn→ℂ)
10	9	a1i 11	. . . 4 ⊢ (𝜑 → exp ∈ (ℂ–cn→ℂ))
11		fveq2 6828	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
12		eqid 2731	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))
13		fvex 6841	. . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V
14	11, 12, 13	fvmpt 6935	. . . . . . 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 2831	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
17	15, 16	eqeltrd 2831	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘) ∈ ℂ)
18	1, 2, 17	serf 13943	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))):𝑍⟶ℂ)
19	1	eqcomi 2740	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) = 𝑍
20	14, 19	eleq2s 2849	. . . . . . 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 13940	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) = seq𝑀( + , 𝐹))
23		climdm 15467	. . . . . 6 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
24	5, 23	sylib 218	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
25	22, 24	eqbrtrd 5115	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
26		climcl 15412	. . . . 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 24826	. . 3 ⊢ (𝜑 → (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
29	11	cbvmptv 5197	. . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))
30	16, 29	fmptd 7053	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)):𝑍⟶ℂ)
31	1, 2, 30	iprodefisumlem 35791	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) = (exp ∘ seq𝑀( + , (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))))
32	1, 2, 3, 4	isum 15632	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))
33	32	fveq2d 6832	. . 3 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝑍 𝐵) = (exp‘( ⇝ ‘seq𝑀( + , 𝐹))))
34	28, 31, 33	3brtr4d 5125	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))) ⇝ (exp‘Σ𝑘 ∈ 𝑍 𝐵))
35		fvco3 6927	. . . 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 6832	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘((𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗))‘𝑘)) = (exp‘(𝐹‘𝑘)))
38	3	fveq2d 6832	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) = (exp‘𝐵))
39	36, 37, 38	3eqtrd 2770	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))‘𝑘) = (exp‘𝐵))
40		efcl 15995	. . 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 15853	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5093 ↦ cmpt 5174 dom cdm 5619 ∘ ccom 5623 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 0cc0 11012 + caddc 11015 · cmul 11017 ℤcz 12474 ℤ_≥cuz 12738 seqcseq 13914 ⇝ cli 15397 Σcsu 15599 ∏cprod 15816 expce 15974 –cn→ccncf 24802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-shft 14980 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-prod 15817 df-ef 15980 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-mulg 18987 df-cntz 19235 df-cmn 19700 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: iprodgam 35793

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