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Mirrors > Home > ILE Home > Th. List > prodf | GIF version |
Description: An infinite product of complex terms is a function from an upper set of integers to ℂ. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
prodf.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
prodf.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
Ref | Expression |
---|---|
prodf | ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodf.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | prodf.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | prodf.3 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
4 | mulcl 7874 | . . 3 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ) | |
5 | 4 | adantl 275 | . 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ) |
6 | 1, 2, 3, 5 | seqf 10390 | 1 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⟶wf 5181 ‘cfv 5185 (class class class)co 5839 ℂcc 7745 · cmul 7752 ℤcz 9185 ℤ≥cuz 9460 seqcseq 10374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-addcom 7847 ax-addass 7849 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-0id 7855 ax-rnegex 7856 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-ltadd 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-frec 6353 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-inn 8852 df-n0 9109 df-z 9186 df-uz 9461 df-seqfrec 10375 |
This theorem is referenced by: clim2prod 11474 clim2divap 11475 prodf1f 11478 prodfap0 11480 prodfrecap 11481 prodfdivap 11482 ntrivcvgap 11483 fproddccvg 11507 fprodseq 11518 |
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