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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 12868 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
3 | ntrivcvgn0.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrrdi 2840 | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
5 | ntrivcvgn0.3 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
6 | climrel 15468 | . . . . 5 ⊢ Rel ⇝ | |
7 | 6 | brrelex2i 5735 | . . . 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 3000 | . . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 0 ↔ 𝑋 ≠ 0)) | |
12 | breq2 5152 | . . . 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 3585 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
15 | seqeq1 14001 | . . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹)) | |
16 | 15 | breq1d 5158 | . . . . 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 1917 | . . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
19 | 18 | rspcev 3609 | . 2 ⊢ ((𝑀 ∈ 𝑍 ∧ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
20 | 4, 14, 19 | syl2anc 583 | 1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 Vcvv 3471 class class class wbr 5148 ‘cfv 6548 0cc0 11138 · cmul 11143 ℤcz 12588 ℤ≥cuz 12852 seqcseq 13998 ⇝ cli 15460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-z 12589 df-uz 12853 df-seq 13999 df-clim 15464 |
This theorem is referenced by: zprodn0 15915 |
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