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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 770 | . . . . 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 10739 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
5 | iccid 12837 | . . . . . . . 8 ⊢ (0 ∈ ℝ* → (0[,]0) = {0}) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]0) = {0} |
7 | 0le0 11788 | . . . . . . . 8 ⊢ 0 ≤ 0 | |
8 | 0le1 11214 | . . . . . . . 8 ⊢ 0 ≤ 1 | |
9 | icccldii 45651 | . . . . . . . 8 ⊢ ((0 ≤ 0 ∧ 0 ≤ 1) → (0[,]0) ∈ (Clsd‘II)) | |
10 | 7, 8, 9 | mp2an 691 | . . . . . . 7 ⊢ (0[,]0) ∈ (Clsd‘II) |
11 | 6, 10 | eqeltrri 2849 | . . . . . 6 ⊢ {0} ∈ (Clsd‘II) |
12 | cnclima 21981 | . . . . . 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 2852 | . . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑆 ∈ (Clsd‘𝐽)) |
15 | simprr 772 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1}))) → 𝑇 = (◡𝑓 “ {1})) | |
16 | 1xr 10751 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
17 | iccid 12837 | . . . . . . . 8 ⊢ (1 ∈ ℝ* → (1[,]1) = {1}) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ (1[,]1) = {1} |
19 | 1le1 11319 | . . . . . . . 8 ⊢ 1 ≤ 1 | |
20 | icccldii 45651 | . . . . . . . 8 ⊢ ((0 ≤ 1 ∧ 1 ≤ 1) → (1[,]1) ∈ (Clsd‘II)) | |
21 | 8, 19, 20 | mp2an 691 | . . . . . . 7 ⊢ (1[,]1) ∈ (Clsd‘II) |
22 | 18, 21 | eqeltrri 2849 | . . . . . 6 ⊢ {1} ∈ (Clsd‘II) |
23 | cnclima 21981 | . . . . . 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 2852 | . . . 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 3205 | . 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 1538 ∈ wcel 2111 ∃wrex 3071 {csn 4525 class class class wbr 5036 ◡ccnv 5527 “ cima 5531 ‘cfv 6340 (class class class)co 7156 0cc0 10588 1c1 10589 ℝ*cxr 10725 ≤ cle 10727 [,]cicc 12795 Clsdccld 21729 Cn ccn 21937 IIcii 23589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-map 8424 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fi 8921 df-sup 8952 df-inf 8953 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-n0 11948 df-z 12034 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-seq 13432 df-exp 13493 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-rest 16767 df-topgen 16788 df-ordt 16845 df-ps 17889 df-tsr 17890 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-top 21607 df-topon 21624 df-bases 21659 df-cld 21732 df-cn 21940 df-ii 23591 |
This theorem is referenced by: (None) |
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