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Mirrors > Home > MPE Home > Th. List > Mathboxes > seppcld | Structured version Visualization version GIF version |
Description: If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.) |
Ref | Expression |
---|---|
seppsepf.1 | ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) |
Ref | Expression |
---|---|
seppcld | ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seppsepf.1 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) | |
2 | simprl 771 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑆 = (◡𝑓 “ {0})) | |
3 | simpl 486 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑓 ∈ (𝐽 Cn II)) | |
4 | 0xr 10904 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
5 | iccid 13004 | . . . . . . . 8 ⊢ (0 ∈ ℝ* → (0[,]0) = {0}) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]0) = {0} |
7 | 0le0 11955 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
8 | 0le1 11379 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
9 | icccldii 45913 | . . . . . . . 8 ⊢ ((0 ≤ 0 ∧ 0 ≤ 1) → (0[,]0) ∈ (Clsd‘II)) | |
10 | 7, 8, 9 | mp2an 692 | . . . . . . 7 ⊢ (0[,]0) ∈ (Clsd‘II) |
11 | 6, 10 | eqeltrri 2836 | . . . . . 6 ⊢ {0} ∈ (Clsd‘II) |
12 | cnclima 22189 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {0} ∈ (Clsd‘II)) → (◡𝑓 “ {0}) ∈ (Clsd‘𝐽)) | |
13 | 3, 11, 12 | sylancl 589 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (◡𝑓 “ {0}) ∈ (Clsd‘𝐽)) |
14 | 2, 13 | eqeltrd 2839 | . . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑆 ∈ (Clsd‘𝐽)) |
15 | simprr 773 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑇 = (◡𝑓 “ {1})) | |
16 | 1xr 10916 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
17 | iccid 13004 | . . . . . . . 8 ⊢ (1 ∈ ℝ* → (1[,]1) = {1}) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (1[,]1) = {1} |
19 | 1le1 11484 | . . . . . . . 8 ⊢ 1 ≤ 1 | |
20 | icccldii 45913 | . . . . . . . 8 ⊢ ((0 ≤ 1 ∧ 1 ≤ 1) → (1[,]1) ∈ (Clsd‘II)) | |
21 | 8, 19, 20 | mp2an 692 | . . . . . . 7 ⊢ (1[,]1) ∈ (Clsd‘II) |
22 | 18, 21 | eqeltrri 2836 | . . . . . 6 ⊢ {1} ∈ (Clsd‘II) |
23 | cnclima 22189 | . . . . . 6 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ {1} ∈ (Clsd‘II)) → (◡𝑓 “ {1}) ∈ (Clsd‘𝐽)) | |
24 | 3, 22, 23 | sylancl 589 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (◡𝑓 “ {1}) ∈ (Clsd‘𝐽)) |
25 | 15, 24 | eqeltrd 2839 | . . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑇 ∈ (Clsd‘𝐽)) |
26 | 14, 25 | jca 515 | . . 3 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
27 | 26 | rexlimiva 3208 | . 2 ⊢ (∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
28 | 1, 27 | syl 17 | 1 ⊢ (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∃wrex 3063 {csn 4555 class class class wbr 5067 ◡ccnv 5564 “ cima 5568 ‘cfv 6397 (class class class)co 7231 0cc0 10753 1c1 10754 ℝ*cxr 10890 ≤ cle 10892 [,]cicc 12962 Clsdccld 21937 Cn ccn 22145 IIcii 23796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fi 9051 df-sup 9082 df-inf 9083 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-n0 12115 df-z 12201 df-uz 12463 df-q 12569 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-ioo 12963 df-ioc 12964 df-ico 12965 df-icc 12966 df-seq 13599 df-exp 13660 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-rest 16951 df-topgen 16972 df-ordt 17030 df-ps 18096 df-tsr 18097 df-psmet 20379 df-xmet 20380 df-met 20381 df-bl 20382 df-mopn 20383 df-top 21815 df-topon 21832 df-bases 21867 df-cld 21940 df-cn 22148 df-ii 23798 |
This theorem is referenced by: (None) |
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