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Mirrors > Home > MPE Home > Th. List > iprodcl | Structured version Visualization version GIF version |
Description: The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.) |
Ref | Expression |
---|---|
iprodcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
iprodcl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iprodcl.3 | ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
iprodcl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
iprodcl.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
iprodcl | ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iprodcl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | iprodcl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | iprodcl.3 | . . 3 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) | |
4 | iprodcl.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
5 | iprodcl.5 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
6 | 1, 2, 3, 4, 5 | iprod 15646 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , 𝐹))) |
7 | fclim 15260 | . . 3 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
8 | 4, 5 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
9 | 1, 3, 8 | ntrivcvg 15607 | . . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ dom ⇝ ) |
10 | ffvelrn 6961 | . . 3 ⊢ (( ⇝ :dom ⇝ ⟶ℂ ∧ seq𝑀( · , 𝐹) ∈ dom ⇝ ) → ( ⇝ ‘seq𝑀( · , 𝐹)) ∈ ℂ) | |
11 | 7, 9, 10 | sylancr 587 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) ∈ ℂ) |
12 | 6, 11 | eqeltrd 2839 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 class class class wbr 5076 dom cdm 5591 ⟶wf 6431 ‘cfv 6435 ℂcc 10867 0cc0 10869 · cmul 10874 ℤcz 12317 ℤ≥cuz 12580 seqcseq 13719 ⇝ cli 15191 ∏cprod 15613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-sup 9199 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12581 df-rp 12729 df-fz 13238 df-fzo 13381 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-prod 15614 |
This theorem is referenced by: (None) |
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