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Mirrors > Home > MPE Home > Th. List > ntrivcvgn0 | Structured version Visualization version GIF version |
Description: A product that converges to a nonzero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.) |
Ref | Expression |
---|---|
ntrivcvgn0.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ntrivcvgn0.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ntrivcvgn0.3 | ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
ntrivcvgn0.4 | ⊢ (𝜑 → 𝑋 ≠ 0) |
Ref | Expression |
---|---|
ntrivcvgn0 | ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrivcvgn0.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 12257 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | ntrivcvgn0.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrrdi 2924 | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | ntrivcvgn0.3 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
7 | climrel 14848 | . . . . 5 ⊢ Rel ⇝ | |
8 | 7 | brrelex2i 5608 | . . . 4 ⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ V) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
10 | ntrivcvgn0.4 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
11 | 10, 6 | jca 514 | . . 3 ⊢ (𝜑 → (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)) |
12 | neeq1 3078 | . . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 0 ↔ 𝑋 ≠ 0)) | |
13 | breq2 5069 | . . . 4 ⊢ (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋)) | |
14 | 12, 13 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝑋 → ((𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋))) |
15 | 9, 11, 14 | spcedv 3598 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
16 | seqeq1 13371 | . . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹)) | |
17 | 16 | breq1d 5075 | . . . . 5 ⊢ (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
18 | 17 | anbi2d 630 | . . . 4 ⊢ (𝑛 = 𝑀 → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
19 | 18 | exbidv 1918 | . . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
20 | 19 | rspcev 3622 | . 2 ⊢ ((𝑀 ∈ 𝑍 ∧ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
21 | 5, 15, 20 | syl2anc 586 | 1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 Vcvv 3494 class class class wbr 5065 ‘cfv 6354 0cc0 10536 · cmul 10541 ℤcz 11980 ℤ≥cuz 12242 seqcseq 13368 ⇝ cli 14840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-pre-lttri 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-neg 10872 df-z 11981 df-uz 12243 df-seq 13369 df-clim 14844 |
This theorem is referenced by: zprodn0 15292 |
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