limcnlp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > limcnlp		Structured version Visualization version GIF version

Theorem limcnlp 25757

Description: If 𝐵 is not a limit point of the domain of the function 𝐹, then every point is a limit of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)

**Hypotheses**
Ref	Expression
limccl.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
limccl.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
limccl.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
ellimc2.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
limcnlp.n	⊢ (𝜑 → ¬ 𝐵 ∈ ((limPt‘𝐾)‘𝐴))

**Assertion**
Ref	Expression
limcnlp	⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ℂ)

Proof of Theorem limcnlp Dummy variables 𝑥 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
2		limccl.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
3		limccl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		ellimc2.k	. . . 4 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
5	1, 2, 3, 4	ellimc2 25756	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝑥 ∈ 𝑢 → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
6	4	cnfldtop 24650	. . . . . . . . . 10 ⊢ 𝐾 ∈ Top
7	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ⊆ ℂ)
8	7	ssdifssd 4137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
9	4	cnfldtopon 24649	. . . . . . . . . . . 12 ⊢ 𝐾 ∈ (TopOn‘ℂ)
10	9	toponunii 22768	. . . . . . . . . . 11 ⊢ ℂ = ∪ 𝐾
11	10	clscld 22901	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∖ {𝐵}) ⊆ ℂ) → ((cls‘𝐾)‘(𝐴 ∖ {𝐵})) ∈ (Clsd‘𝐾))
12	6, 8, 11	sylancr 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((cls‘𝐾)‘(𝐴 ∖ {𝐵})) ∈ (Clsd‘𝐾))
13	10	cldopn 22885	. . . . . . . . 9 ⊢ (((cls‘𝐾)‘(𝐴 ∖ {𝐵})) ∈ (Clsd‘𝐾) → (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∈ 𝐾)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∈ 𝐾)
15		limcnlp.n	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐵 ∈ ((limPt‘𝐾)‘𝐴))
16	10	islp 22994	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))))
17	6, 2, 16	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐾)‘𝐴) ↔ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))))
18	15, 17	mtbid 324	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐵 ∈ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))
19	3, 18	eldifd 3954	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))))
21		difin2 4286	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝐵}) ⊆ ℂ → ((𝐴 ∖ {𝐵}) ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) = ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵})))
22	8, 21	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴 ∖ {𝐵}) ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) = ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵})))
23	10	sscls 22910	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∖ {𝐵}) ⊆ ℂ) → (𝐴 ∖ {𝐵}) ⊆ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))
24	6, 8, 23	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴 ∖ {𝐵}) ⊆ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))
25		ssdif0 4358	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝐵}) ⊆ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})) ↔ ((𝐴 ∖ {𝐵}) ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) = ∅)
26	24, 25	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝐴 ∖ {𝐵}) ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) = ∅)
27	22, 26	eqtr3d 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵})) = ∅)
28	27	imaeq2d 6052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ∅))
29		ima0 6069	. . . . . . . . . 10 ⊢ (𝐹 “ ∅) = ∅
30	28, 29	eqtrdi 2782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) = ∅)
31		0ss 4391	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑢
32	30, 31	eqsstrdi 4031	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)
33		eleq2 2816	. . . . . . . . . 10 ⊢ (𝑣 = (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵})))))
34		ineq1 4200	. . . . . . . . . . . 12 ⊢ (𝑣 = (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) → (𝑣 ∩ (𝐴 ∖ {𝐵})) = ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵})))
35	34	imaeq2d 6052	. . . . . . . . . . 11 ⊢ (𝑣 = (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) → (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))))
36	35	sseq1d 4008	. . . . . . . . . 10 ⊢ (𝑣 = (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) → ((𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
37	33, 36	anbi12d 630	. . . . . . . . 9 ⊢ (𝑣 = (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) → ((𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵 ∈ (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∧ (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
38	37	rspcev 3606	. . . . . . . 8 ⊢ (((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∈ 𝐾 ∧ (𝐵 ∈ (ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∧ (𝐹 “ ((ℂ ∖ ((cls‘𝐾)‘(𝐴 ∖ {𝐵}))) ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
39	14, 20, 32, 38	syl12anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
40	39	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ 𝑢 → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
41	40	ralrimivw 3144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∀𝑢 ∈ 𝐾 (𝑥 ∈ 𝑢 → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
42	41	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ → ∀𝑢 ∈ 𝐾 (𝑥 ∈ 𝑢 → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
43	42	pm4.71d 561	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℂ ↔ (𝑥 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝑥 ∈ 𝑢 → ∃𝑣 ∈ 𝐾 (𝐵 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
44	5, 43	bitr4d 282	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹 lim_ℂ 𝐵) ↔ 𝑥 ∈ ℂ))
45	44	eqrdv 2724	1 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ℂ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ∖ cdif 3940 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 {csn 4623 “ cima 5672 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 TopOpenctopn 17373 ℂ_fldccnfld 21235 Topctop 22745 Clsdccld 22870 clsccl 22872 limPtclp 22988 lim_ℂ climc 25741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-cls 22875 df-lp 22990 df-cnp 23082 df-xms 24176 df-ms 24177 df-limc 25745

This theorem is referenced by: (None)

W3C validator