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 17048 | . . . . 5 ⊢ ℝ* = ran ≤ | |
2 | df-rn 5049 | . . . . 5 ⊢ ran ≤ = dom ◡ ≤ | |
3 | 1, 2 | eqtri 2632 | . . . 4 ⊢ ℝ* = dom ◡ ≤ |
4 | letsr 17050 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
5 | cnvtsr 17045 | . . . . . 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 5225 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) | |
11 | 10 | adantlr 747 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) |
12 | simpr 476 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑧 ∈ ℝ*) | |
13 | simplr 788 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑥 ∈ 𝐴) | |
14 | brcnvg 5225 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 ∈ 𝐴) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) | |
15 | 12, 13, 14 | syl2anc 691 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) |
16 | 11, 15 | anbi12d 743 | . . . . . . . 8 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧))) |
17 | ancom 465 | . . . . . . . 8 ⊢ ((𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | |
18 | 16, 17 | syl6bb 275 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) |
19 | 18 | rabbidva 3163 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
20 | simpr 476 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
21 | 8, 20 | sseldi 3566 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
22 | simpl 472 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
23 | 8, 22 | sseldi 3566 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
24 | iccval 12085 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
25 | 21, 23, 24 | syl2anc 691 | . . . . . . 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 468 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
28 | 25, 27 | eqsstr3d 3603 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
29 | 19, 28 | eqsstrd 3602 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
30 | 29 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
31 | 3, 7, 9, 30 | ordtrest2 20818 | . . 3 ⊢ (⊤ → (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴)) |
32 | 31 | trud 1484 | . 2 ⊢ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴) |
33 | tsrps 17044 | . . . . 5 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel) | |
34 | 4, 33 | ax-mp 5 | . . . 4 ⊢ ≤ ∈ PosetRel |
35 | ordtcnv 20815 | . . . 4 ⊢ ( ≤ ∈ PosetRel → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | |
36 | 34, 35 | ax-mp 5 | . . 3 ⊢ (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ) |
37 | 36 | oveq1i 6559 | . 2 ⊢ ((ordTop‘◡ ≤ ) ↾t 𝐴) = ((ordTop‘ ≤ ) ↾t 𝐴) |
38 | 32, 37 | eqtr2i 2633 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 ↾t crest 15904 ordTopcordt 15982 PosetRelcps 17021 TosetRel ctsr 17022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 df-rest 15906 df-topgen 15927 df-ordt 15984 df-ps 17023 df-tsr 17024 df-top 20521 df-bases 20522 df-topon 20523 |
This theorem is referenced by: xrge0iifhmeo 29310 |
Copyright terms: Public domain | W3C validator |