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 17835 | . . . . 5 ⊢ ℝ* = ran ≤ | |
2 | df-rn 5566 | . . . . 5 ⊢ ran ≤ = dom ◡ ≤ | |
3 | 1, 2 | eqtri 2844 | . . . 4 ⊢ ℝ* = dom ◡ ≤ |
4 | letsr 17837 | . . . . . 6 ⊢ ≤ ∈ TosetRel | |
5 | cnvtsr 17832 | . . . . . 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 5750 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) | |
11 | 10 | adantlr 713 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)) |
12 | simpr 487 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑧 ∈ ℝ*) | |
13 | simplr 767 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → 𝑥 ∈ 𝐴) | |
14 | brcnvg 5750 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ ℝ* ∧ 𝑥 ∈ 𝐴) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) | |
15 | 12, 13, 14 | syl2anc 586 | . . . . . . . . 9 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)) |
16 | 11, 15 | anbi12d 632 | . . . . . . . 8 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧))) |
17 | ancom 463 | . . . . . . . 8 ⊢ ((𝑧 ≤ 𝑦 ∧ 𝑥 ≤ 𝑧) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | |
18 | 16, 17 | syl6bb 289 | . . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ℝ*) → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) |
19 | 18 | rabbidva 3478 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
20 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
21 | 8, 20 | sseldi 3965 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
22 | simpl 485 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
23 | 8, 22 | sseldi 3965 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ*) |
24 | iccval 12778 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
25 | 21, 23, 24 | syl2anc 586 | . . . . . . 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 461 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) |
28 | 25, 27 | eqsstrrd 4006 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) |
29 | 19, 28 | eqsstrd 4005 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
30 | 29 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴) |
31 | 3, 7, 9, 30 | ordtrest2 21812 | . . 3 ⊢ (⊤ → (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴)) |
32 | 31 | mptru 1544 | . 2 ⊢ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘◡ ≤ ) ↾t 𝐴) |
33 | tsrps 17831 | . . . . 5 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel) | |
34 | 4, 33 | ax-mp 5 | . . . 4 ⊢ ≤ ∈ PosetRel |
35 | ordtcnv 21809 | . . . 4 ⊢ ( ≤ ∈ PosetRel → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | |
36 | 34, 35 | ax-mp 5 | . . 3 ⊢ (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ) |
37 | 36 | oveq1i 7166 | . 2 ⊢ ((ordTop‘◡ ≤ ) ↾t 𝐴) = ((ordTop‘ ≤ ) ↾t 𝐴) |
38 | 32, 37 | eqtr2i 2845 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 {crab 3142 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 × cxp 5553 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ℝ*cxr 10674 ≤ cle 10676 [,]cicc 12742 ↾t crest 16694 ordTopcordt 16772 PosetRelcps 17808 TosetRel ctsr 17809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-icc 12746 df-rest 16696 df-topgen 16717 df-ordt 16774 df-ps 17810 df-tsr 17811 df-top 21502 df-topon 21519 df-bases 21554 |
This theorem is referenced by: xrge0iifhmeo 31179 |
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