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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 12598 | . . 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 15201 | . . . . 5 ⊢ Rel ⇝ | |
7 | 6 | brrelex2i 5644 | . . . 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 512 | . . 3 ⊢ (𝜑 → (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋)) |
11 | neeq1 3006 | . . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 0 ↔ 𝑋 ≠ 0)) | |
12 | breq2 5078 | . . . 4 ⊢ (𝑦 = 𝑋 → (seq𝑀( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑋)) | |
13 | 11, 12 | anbi12d 631 | . . 3 ⊢ (𝑦 = 𝑋 → ((𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦) ↔ (𝑋 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑋))) |
14 | 8, 10, 13 | spcedv 3537 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
15 | seqeq1 13724 | . . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹)) | |
16 | 15 | breq1d 5084 | . . . . 5 ⊢ (𝑛 = 𝑀 → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
17 | 16 | anbi2d 629 | . . . 4 ⊢ (𝑛 = 𝑀 → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
18 | 17 | exbidv 1924 | . . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
19 | 18 | rspcev 3561 | . 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 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 class class class wbr 5074 ‘cfv 6433 0cc0 10871 · cmul 10876 ℤcz 12319 ℤ≥cuz 12582 seqcseq 13721 ⇝ cli 15193 |
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-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 |
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-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 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-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-neg 11208 df-z 12320 df-uz 12583 df-seq 13722 df-clim 15197 |
This theorem is referenced by: zprodn0 15649 |
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