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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmlim | Structured version Visualization version GIF version |
Description: Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on ℂ on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
Ref | Expression |
---|---|
lmlim.j | ⊢ 𝐽 ∈ (TopOn‘𝑌) |
lmlim.f | ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) |
lmlim.p | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
lmlim.t | ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
lmlim.x | ⊢ 𝑋 ⊆ ℂ |
Ref | Expression |
---|---|
lmlim | ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (𝐽 ↾t 𝑋) | |
2 | nnuz 12029 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | cnex 10353 | . . . . 5 ⊢ ℂ ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → ℂ ∈ V) |
5 | lmlim.x | . . . . 5 ⊢ 𝑋 ⊆ ℂ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
7 | 4, 6 | ssexd 5042 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | lmlim.j | . . . . 5 ⊢ 𝐽 ∈ (TopOn‘𝑌) | |
9 | 8 | topontopi 21127 | . . . 4 ⊢ 𝐽 ∈ Top |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
11 | lmlim.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | 1z 11759 | . . . 4 ⊢ 1 ∈ ℤ | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) |
14 | lmlim.f | . . 3 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
15 | 1, 2, 7, 10, 11, 13, 14 | lmss 21510 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹(⇝𝑡‘(𝐽 ↾t 𝑋))𝑃)) |
16 | lmlim.t | . . . . 5 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) | |
17 | 16 | fveq2i 6449 | . . . 4 ⊢ (⇝𝑡‘(𝐽 ↾t 𝑋)) = (⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋)) |
18 | 17 | breqi 4892 | . . 3 ⊢ (𝐹(⇝𝑡‘(𝐽 ↾t 𝑋))𝑃 ↔ 𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃) |
19 | 18 | a1i 11 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘(𝐽 ↾t 𝑋))𝑃 ↔ 𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃)) |
20 | eqid 2778 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) | |
21 | eqid 2778 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | 21 | cnfldtop 22995 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
24 | 20, 2, 7, 23, 11, 13, 14 | lmss 21510 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃)) |
25 | fss 6304 | . . . . 5 ⊢ ((𝐹:ℕ⟶𝑋 ∧ 𝑋 ⊆ ℂ) → 𝐹:ℕ⟶ℂ) | |
26 | 14, 5, 25 | sylancl 580 | . . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ) |
27 | 21, 2 | lmclimf 23510 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝐹:ℕ⟶ℂ) → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
28 | 12, 26, 27 | sylancr 581 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
29 | 24, 28 | bitr3d 273 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
30 | 15, 19, 29 | 3bitrd 297 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 1c1 10273 ℕcn 11374 ℤcz 11728 ⇝ cli 14623 ↾t crest 16467 TopOpenctopn 16468 ℂfldccnfld 20142 Topctop 21105 TopOnctopon 21122 ⇝𝑡clm 21438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-lm 21441 df-xms 22533 df-ms 22534 |
This theorem is referenced by: lmlimxrge0 30592 |
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