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Mirrors > Home > MPE Home > Th. List > zprodn0 | Structured version Visualization version GIF version |
Description: Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.) |
Ref | Expression |
---|---|
zprodn0.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
zprodn0.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
zprodn0.3 | ⊢ (𝜑 → 𝑋 ≠ 0) |
zprodn0.4 | ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
zprodn0.5 | ⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
zprodn0.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
zprodn0.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
zprodn0 | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zprodn0.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | zprodn0.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | zprodn0.4 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
4 | zprodn0.3 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
5 | 1, 2, 3, 4 | ntrivcvgn0 15610 | . . 3 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑥(𝑥 ≠ 0 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) |
6 | zprodn0.5 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
7 | zprodn0.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) | |
8 | zprodn0.7 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
9 | 1, 2, 5, 6, 7, 8 | zprod 15647 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹))) |
10 | fclim 15262 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
11 | ffun 6603 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
13 | funbrfv 6820 | . . 3 ⊢ (Fun ⇝ → (seq𝑀( · , 𝐹) ⇝ 𝑋 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋)) | |
14 | 12, 3, 13 | mpsyl 68 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
15 | 9, 14 | eqtrd 2778 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 dom cdm 5589 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 ℤcz 12319 ℤ≥cuz 12582 seqcseq 13721 ⇝ cli 15193 ∏cprod 15615 |
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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
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 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616 |
This theorem is referenced by: iprodn0 15650 prod0 15653 prod1 15654 |
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