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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 2821 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑃 ∈ 𝑢 ↔ 𝑃 ∈ 𝑈)) | |
3 | eleq2 2821 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ 𝑈)) | |
4 | 3 | rexralbidv 3211 | . . . 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 21982 | . . . . . . . . 9 ⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) | |
8 | 6, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top) |
9 | toptopon2 21669 | . . . . . . . 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 22010 | . . . . . 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 1145 | . . . 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 3075 | . . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) |
18 | 17 | reximi 3157 | . . . . . 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 3075 | . . . 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 3528 | . 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 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ∪ cuni 4796 class class class wbr 5030 dom cdm 5525 ‘cfv 6339 (class class class)co 7170 ↑pm cpm 8438 ℂcc 10613 ℤcz 12062 ℤ≥cuz 12324 Topctop 21644 TopOnctopon 21661 ⇝𝑡clm 21977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-pre-lttri 10689 ax-pre-lttrn 10690 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-er 8320 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-neg 10951 df-z 12063 df-uz 12325 df-top 21645 df-topon 21662 df-lm 21980 |
This theorem is referenced by: lmmo 22131 1stccnp 22213 1stckgenlem 22304 iscmet3lem2 24044 |
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