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 2821 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (𝐽 ↾t 𝑋) | |
2 | nnuz 12275 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | cnex 10612 | . . . . 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 5220 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | lmlim.j | . . . . 5 ⊢ 𝐽 ∈ (TopOn‘𝑌) | |
9 | 8 | topontopi 21517 | . . . 4 ⊢ 𝐽 ∈ Top |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
11 | lmlim.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | 1z 12006 | . . . 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 21900 | . 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 6667 | . . . 4 ⊢ (⇝𝑡‘(𝐽 ↾t 𝑋)) = (⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋)) |
18 | 17 | breqi 5064 | . . 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 2821 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) | |
21 | eqid 2821 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | 21 | cnfldtop 23386 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ Top) |
24 | 20, 2, 7, 23, 11, 13, 14 | lmss 21900 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃)) |
25 | fss 6521 | . . . . 5 ⊢ ((𝐹:ℕ⟶𝑋 ∧ 𝑋 ⊆ ℂ) → 𝐹:ℕ⟶ℂ) | |
26 | 14, 5, 25 | sylancl 588 | . . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ) |
27 | 21, 2 | lmclimf 23901 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝐹:ℕ⟶ℂ) → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
28 | 12, 26, 27 | sylancr 589 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑡‘(TopOpen‘ℂfld))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
29 | 24, 28 | bitr3d 283 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘((TopOpen‘ℂfld) ↾t 𝑋))𝑃 ↔ 𝐹 ⇝ 𝑃)) |
30 | 15, 19, 29 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 ℕcn 11632 ℤcz 11975 ⇝ cli 14835 ↾t crest 16688 TopOpenctopn 16689 ℂfldccnfld 20539 Topctop 21495 TopOnctopon 21512 ⇝𝑡clm 21828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-rest 16690 df-topn 16691 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-lm 21831 df-xms 22924 df-ms 22925 |
This theorem is referenced by: lmlimxrge0 31186 |
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