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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 12261 | . . . 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 2926 | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | ntrivcvgn0.3 | . . . 4 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) | |
7 | climrel 14851 | . . . . 5 ⊢ Rel ⇝ | |
8 | 7 | brrelex2i 5611 | . . . 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 3080 | . . . 4 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 0 ↔ 𝑋 ≠ 0)) | |
13 | breq2 5072 | . . . 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 3601 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦)) |
16 | seqeq1 13375 | . . . . . 6 ⊢ (𝑛 = 𝑀 → seq𝑛( · , 𝐹) = seq𝑀( · , 𝐹)) | |
17 | 16 | breq1d 5078 | . . . . 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 1922 | . . 3 ⊢ (𝑛 = 𝑀 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ 𝑦))) |
20 | 19 | rspcev 3625 | . 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 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 0cc0 10539 · cmul 10544 ℤcz 11984 ℤ≥cuz 12246 seqcseq 13372 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-neg 10875 df-z 11985 df-uz 12247 df-seq 13373 df-clim 14847 |
This theorem is referenced by: zprodn0 15295 |
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