![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 17611 | . . . . 5 ⊢ ℝ* = ran ≤ | |
2 | df-rn 5366 | . . . . 5 ⊢ ran ≤ = dom ◡ ≤ | |
3 | 1, 2 | eqtri 2802 | . . . 4 ⊢ ℝ* = dom ◡ ≤ |
4 | letsr 17613 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
5 | cnvtsr 17608 | . . . . . 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 5547 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) | |
11 | 10 | adantlr 705 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) |
12 | simpr 479 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑧 ∈ ℝ*) | |
13 | simplr 759 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑥 ∈ 𝐴) | |
14 | brcnvg 5547 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 ∈ 𝐴) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) | |
15 | 12, 13, 14 | syl2anc 579 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) |
16 | 11, 15 | anbi12d 624 | . . . . . . . 8 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧))) |
17 | ancom 454 | . . . . . . . 8 ⊢ ((𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | |
18 | 16, 17 | syl6bb 279 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) |
19 | 18 | rabbidva 3385 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
20 | simpr 479 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
21 | 8, 20 | sseldi 3819 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
22 | simpl 476 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
23 | 8, 22 | sseldi 3819 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
24 | iccval 12526 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
25 | 21, 23, 24 | syl2anc 579 | . . . . . . 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 452 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
28 | 25, 27 | eqsstr3d 3859 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
29 | 19, 28 | eqsstrd 3858 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
30 | 29 | adantl 475 | . . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
31 | 3, 7, 9, 30 | ordtrest2 21416 | . . 3 ⊢ (⊤ → (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴)) |
32 | 31 | mptru 1609 | . 2 ⊢ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴) |
33 | tsrps 17607 | . . . . 5 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel) | |
34 | 4, 33 | ax-mp 5 | . . . 4 ⊢ ≤ ∈ PosetRel |
35 | ordtcnv 21413 | . . . 4 ⊢ ( ≤ ∈ PosetRel → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | |
36 | 34, 35 | ax-mp 5 | . . 3 ⊢ (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ) |
37 | 36 | oveq1i 6932 | . 2 ⊢ ((ordTop‘◡ ≤ ) ↾t 𝐴) = ((ordTop‘ ≤ ) ↾t 𝐴) |
38 | 32, 37 | eqtr2i 2803 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ⊤wtru 1602 ∈ wcel 2107 {crab 3094 ∩ cin 3791 ⊆ wss 3792 class class class wbr 4886 × cxp 5353 ◡ccnv 5354 dom cdm 5355 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ℝ*cxr 10410 ≤ cle 10412 [,]cicc 12490 ↾t crest 16467 ordTopcordt 16545 PosetRelcps 17584 TosetRel ctsr 17585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-icc 12494 df-rest 16469 df-topgen 16490 df-ordt 16547 df-ps 17586 df-tsr 17587 df-top 21106 df-topon 21123 df-bases 21158 |
This theorem is referenced by: xrge0iifhmeo 30580 |
Copyright terms: Public domain | W3C validator |