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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 | 1 | uzidd 12892 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | ntrivcvgn0.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrrdi 2850 | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
5 | ntrivcvgn0.3 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
6 | climrel 15525 | . . . . 5 ⊢ Rel ⇝ | |
7 | 6 | brrelex2i 5746 | . . . 4 ⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ V) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
9 | ntrivcvgn0.4 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
10 | 9, 5 | jca 511 | . . 3 ⊢ (𝜑 → (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)) |
11 | neeq1 3001 | . . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 0 ↔ 𝑋 ≠ 0)) | |
12 | breq2 5152 | . . . 4 ⊢ (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋)) | |
13 | 11, 12 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝑋 → ((𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋))) |
14 | 8, 10, 13 | spcedv 3598 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
15 | seqeq1 14042 | . . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹)) | |
16 | 15 | breq1d 5158 | . . . . 5 ⊢ (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
17 | 16 | anbi2d 630 | . . . 4 ⊢ (𝑛 = 𝑀 → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
18 | 17 | exbidv 1919 | . . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
19 | 18 | rspcev 3622 | . 2 ⊢ ((𝑀 ∈ 𝑍 ∧ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
20 | 4, 14, 19 | syl2anc 584 | 1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 0cc0 11153 · cmul 11158 ℤcz 12611 ℤ≥cuz 12876 seqcseq 14039 ⇝ cli 15517 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-seq 14040 df-clim 15521 |
This theorem is referenced by: zprodn0 15972 |
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