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Mirrors > Home > MPE Home > Th. List > ordtrestixx | Structured version Visualization version GIF version |
Description: The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
ordtrestixx.1 | ⊢ 𝐴 ⊆ ℝ* |
ordtrestixx.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
Ref | Expression |
---|---|
ordtrestixx | ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ledm 17047 | . . . 4 ⊢ ℝ* = dom ≤ | |
2 | letsr 17050 | . . . . 5 ⊢ ≤ ∈ TosetRel | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ≤ ∈ TosetRel ) |
4 | ordtrestixx.1 | . . . . 5 ⊢ 𝐴 ⊆ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ*) |
6 | 4 | sseli 3564 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ*) |
7 | 4 | sseli 3564 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ*) |
8 | iccval 12085 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
9 | 6, 7, 8 | syl2an 493 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
10 | ordtrestixx.2 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) | |
11 | 9, 10 | eqsstr3d 3603 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
13 | 1, 3, 5, 12 | ordtrest2 20818 | . . 3 ⊢ (⊤ → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴)) |
14 | 13 | eqcomd 2616 | . 2 ⊢ (⊤ → ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) |
15 | 14 | trud 1484 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 ≤ cle 9954 [,]cicc 12049 ↾t crest 15904 ordTopcordt 15982 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: ordtresticc 20837 icopnfhmeo 22550 |
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