Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvordtrestixx | Structured version Visualization version GIF version |
Description: The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
Ref | Expression |
---|---|
cnvordtrestixx.1 | ⊢ 𝐴 ⊆ ℝ* |
cnvordtrestixx.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
Ref | Expression |
---|---|
cnvordtrestixx | ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lern 17823 | . . . . 5 ⊢ ℝ* = ran ≤ | |
2 | df-rn 5559 | . . . . 5 ⊢ ran ≤ = dom ◡ ≤ | |
3 | 1, 2 | eqtri 2841 | . . . 4 ⊢ ℝ* = dom ◡ ≤ |
4 | letsr 17825 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
5 | cnvtsr 17820 | . . . . . 6 ⊢ ( ≤ ∈ TosetRel → ◡ ≤ ∈ TosetRel ) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ ◡ ≤ ∈ TosetRel |
7 | 6 | a1i 11 | . . . 4 ⊢ (⊤ → ◡ ≤ ∈ TosetRel ) |
8 | cnvordtrestixx.1 | . . . . 5 ⊢ 𝐴 ⊆ ℝ* | |
9 | 8 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ*) |
10 | brcnvg 5743 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) | |
11 | 10 | adantlr 711 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) |
12 | simpr 485 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑧 ∈ ℝ*) | |
13 | simplr 765 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑥 ∈ 𝐴) | |
14 | brcnvg 5743 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 ∈ 𝐴) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) | |
15 | 12, 13, 14 | syl2anc 584 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) |
16 | 11, 15 | anbi12d 630 | . . . . . . . 8 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧))) |
17 | ancom 461 | . . . . . . . 8 ⊢ ((𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | |
18 | 16, 17 | syl6bb 288 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) |
19 | 18 | rabbidva 3476 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
20 | simpr 485 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
21 | 8, 20 | sseldi 3962 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
22 | simpl 483 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
23 | 8, 22 | sseldi 3962 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
24 | iccval 12765 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
25 | 21, 23, 24 | syl2anc 584 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
26 | cnvordtrestixx.2 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) | |
27 | 26 | ancoms 459 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
28 | 25, 27 | eqsstrrd 4003 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
29 | 19, 28 | eqsstrd 4002 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
30 | 29 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
31 | 3, 7, 9, 30 | ordtrest2 21740 | . . 3 ⊢ (⊤ → (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴)) |
32 | 31 | mptru 1535 | . 2 ⊢ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴) |
33 | tsrps 17819 | . . . . 5 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel) | |
34 | 4, 33 | ax-mp 5 | . . . 4 ⊢ ≤ ∈ PosetRel |
35 | ordtcnv 21737 | . . . 4 ⊢ ( ≤ ∈ PosetRel → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | |
36 | 34, 35 | ax-mp 5 | . . 3 ⊢ (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ) |
37 | 36 | oveq1i 7155 | . 2 ⊢ ((ordTop‘◡ ≤ ) ↾t 𝐴) = ((ordTop‘ ≤ ) ↾t 𝐴) |
38 | 32, 37 | eqtr2i 2842 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 {crab 3139 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 × cxp 5546 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℝ*cxr 10662 ≤ cle 10664 [,]cicc 12729 ↾t crest 16682 ordTopcordt 16760 PosetRelcps 17796 TosetRel ctsr 17797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-icc 12733 df-rest 16684 df-topgen 16705 df-ordt 16762 df-ps 17798 df-tsr 17799 df-top 21430 df-topon 21447 df-bases 21482 |
This theorem is referenced by: xrge0iifhmeo 31078 |
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