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Mirrors > Home > MPE Home > Th. List > limcmpt | Structured version Visualization version GIF version |
Description: Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
limcmpt.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
limcmpt.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
limcmpt.f | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℂ) |
limcmpt.j | ⊢ 𝐽 = (𝐾 ↾t (𝐴 ∪ {𝐵})) |
limcmpt.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
limcmpt | ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, 𝐷)) ∈ ((𝐽 CnP 𝐾)‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcmpt.j | . . 3 ⊢ 𝐽 = (𝐾 ↾t (𝐴 ∪ {𝐵})) | |
2 | limcmpt.k | . . 3 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
3 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑦if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧)) | |
4 | nfv 1910 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 = 𝐵 | |
5 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
6 | nffvmpt1 6903 | . . . . 5 ⊢ Ⅎ𝑧((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑦) | |
7 | 4, 5, 6 | nfif 4555 | . . . 4 ⊢ Ⅎ𝑧if(𝑦 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑦)) |
8 | eqeq1 2732 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑦 = 𝐵)) | |
9 | fveq2 6892 | . . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑦)) | |
10 | 8, 9 | ifbieq2d 4551 | . . . 4 ⊢ (𝑧 = 𝑦 → if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧)) = if(𝑦 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑦))) |
11 | 3, 7, 10 | cbvmpt 5254 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧))) = (𝑦 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑦 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑦))) |
12 | limcmpt.f | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐷 ∈ ℂ) | |
13 | 12 | fmpttd 7120 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐷):𝐴⟶ℂ) |
14 | limcmpt.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
15 | limcmpt.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
16 | 1, 2, 11, 13, 14, 15 | ellimc 25796 | . 2 ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵))) |
17 | elun 4145 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵})) | |
18 | velsn 4641 | . . . . . . . . . 10 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵) | |
19 | 18 | orbi2i 911 | . . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) |
20 | 17, 19 | bitri 275 | . . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) |
21 | pm5.61 999 | . . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) | |
22 | 21 | simplbi 497 | . . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ 𝐴) |
23 | 20, 22 | sylanb 580 | . . . . . . 7 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑧 = 𝐵) → 𝑧 ∈ 𝐴) |
24 | 23, 12 | sylan2 592 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑧 = 𝐵)) → 𝐷 ∈ ℂ) |
25 | eqid 2728 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐷) = (𝑧 ∈ 𝐴 ↦ 𝐷) | |
26 | 25 | fvmpt2 7011 | . . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐷 ∈ ℂ) → ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧) = 𝐷) |
27 | 23, 24, 26 | syl2an2 685 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑧 = 𝐵)) → ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧) = 𝐷) |
28 | 27 | anassrs 467 | . . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑧 = 𝐵) → ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧) = 𝐷) |
29 | 28 | ifeq2da 4557 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧)) = if(𝑧 = 𝐵, 𝐶, 𝐷)) |
30 | 29 | mpteq2dva 5243 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, 𝐷))) |
31 | 30 | eleq1d 2814 | . 2 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, ((𝑧 ∈ 𝐴 ↦ 𝐷)‘𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, 𝐷)) ∈ ((𝐽 CnP 𝐾)‘𝐵))) |
32 | 16, 31 | bitrd 279 | 1 ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ 𝐴 ↦ 𝐷) limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, 𝐷)) ∈ ((𝐽 CnP 𝐾)‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∪ cun 3943 ⊆ wss 3945 ifcif 4525 {csn 4625 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7415 ℂcc 11131 ↾t crest 17396 TopOpenctopn 17397 ℂfldccnfld 21273 CnP ccnp 23123 limℂ climc 25785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fi 9429 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-fz 13512 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17398 df-topn 17399 df-topgen 17419 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cnp 23126 df-xms 24220 df-ms 24221 df-limc 25789 |
This theorem is referenced by: limcmpt2 25807 limccnp2 25815 limcco 25816 |
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