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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 15620 | . . 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 15657 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹))) |
10 | fclim 15272 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
11 | ffun 6595 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
12 | 10, 11 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
13 | funbrfv 6812 | . . 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 3886 ifcif 4459 class class class wbr 5073 dom cdm 5584 Fun wfun 6420 ⟶wf 6422 ‘cfv 6426 ℂcc 10879 0cc0 10881 1c1 10882 · cmul 10886 ℤcz 12329 ℤ≥cuz 12592 seqcseq 13731 ⇝ cli 15203 ∏cprod 15625 |
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 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
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-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 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 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-sup 9188 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-rp 12741 df-fz 13250 df-fzo 13393 df-seq 13732 df-exp 13793 df-hash 14055 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-clim 15207 df-prod 15626 |
This theorem is referenced by: iprodn0 15660 prod0 15663 prod1 15664 |
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