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Mirrors > Home > MPE Home > Th. List > lmcvg | Structured version Visualization version GIF version |
Description: Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
lmcvg.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
lmcvg.3 | ⊢ (𝜑 → 𝑃 ∈ 𝑈) |
lmcvg.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
lmcvg.5 | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) |
lmcvg.6 | ⊢ (𝜑 → 𝑈 ∈ 𝐽) |
Ref | Expression |
---|---|
lmcvg | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmcvg.3 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑈) | |
2 | eleq2 2878 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑃 ∈ 𝑢 ↔ 𝑃 ∈ 𝑈)) | |
3 | eleq2 2878 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ 𝑈)) | |
4 | 3 | rexralbidv 3260 | . . . 4 ⊢ (𝑢 = 𝑈 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈)) |
5 | 2, 4 | imbi12d 348 | . . 3 ⊢ (𝑢 = 𝑈 → ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ↔ (𝑃 ∈ 𝑈 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈))) |
6 | lmcvg.5 | . . . . . 6 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | |
7 | lmrcl 21836 | . . . . . . . . 9 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) | |
8 | 6, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top) |
9 | toptopon2 21523 | . . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
10 | 8, 9 | sylib 221 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | lmcvg.1 | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
12 | lmcvg.4 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 10, 11, 12 | lmbr2 21864 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑃 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
14 | 6, 13 | mpbid 235 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝐽 ↑pm ℂ) ∧ 𝑃 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
15 | 14 | simp3d 1141 | . . . 4 ⊢ (𝜑 → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
16 | simpr 488 | . . . . . . . 8 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → (𝐹‘𝑘) ∈ 𝑢) | |
17 | 16 | ralimi 3128 | . . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) |
18 | 17 | reximi 3206 | . . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) |
19 | 18 | imim2i 16 | . . . . 5 ⊢ ((𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) → (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) |
20 | 19 | ralimi 3128 | . . . 4 ⊢ (∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) |
21 | 15, 20 | syl 17 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) |
22 | lmcvg.6 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐽) | |
23 | 5, 21, 22 | rspcdva 3573 | . 2 ⊢ (𝜑 → (𝑃 ∈ 𝑈 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈)) |
24 | 1, 23 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 ℤcz 11969 ℤ≥cuz 12231 Topctop 21498 TopOnctopon 21515 ⇝𝑡clm 21831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-top 21499 df-topon 21516 df-lm 21834 |
This theorem is referenced by: lmmo 21985 1stccnp 22067 1stckgenlem 22158 iscmet3lem2 23896 |
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