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Mirrors > Home > MPE Home > Th. List > leordtval | Structured version Visualization version GIF version |
Description: The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
leordtval.1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) |
leordtval.2 | ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) |
leordtval.3 | ⊢ 𝐶 = ran (,) |
Ref | Expression |
---|---|
leordtval | ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leordtval.1 | . . 3 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) | |
2 | leordtval.2 | . . 3 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) | |
3 | 1, 2 | leordtval2 22707 | . 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵))) |
4 | letsr 18542 | . . . 4 ⊢ ≤ ∈ TosetRel | |
5 | ledm 18539 | . . . . 5 ⊢ ℝ* = dom ≤ | |
6 | 1 | leordtvallem1 22705 | . . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
7 | 1, 2 | leordtvallem2 22706 | . . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
8 | leordtval.3 | . . . . . 6 ⊢ 𝐶 = ran (,) | |
9 | df-ioo 13324 | . . . . . . . 8 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)}) | |
10 | xrltnle 11277 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎)) | |
11 | 10 | adantlr 713 | . . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎)) |
12 | xrltnle 11277 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) | |
13 | 12 | ancoms 459 | . . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) |
14 | 13 | adantll 712 | . . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) |
15 | 11, 14 | anbi12d 631 | . . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → ((𝑎 < 𝑦 ∧ 𝑦 < 𝑏) ↔ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦))) |
16 | 15 | rabbidva 3439 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → {𝑦 ∈ ℝ* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)} = {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) |
17 | 16 | mpoeq3ia 7483 | . . . . . . . 8 ⊢ (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)}) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) |
18 | 9, 17 | eqtri 2760 | . . . . . . 7 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) |
19 | 18 | rneqi 5934 | . . . . . 6 ⊢ ran (,) = ran (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) |
20 | 8, 19 | eqtri 2760 | . . . . 5 ⊢ 𝐶 = ran (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) |
21 | 5, 6, 7, 20 | ordtbas2 22686 | . . . 4 ⊢ ( ≤ ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) |
22 | 4, 21 | ax-mp 5 | . . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶) |
23 | 22 | fveq2i 6891 | . 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) |
24 | 3, 23 | eqtri 2760 | 1 ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 ∪ cun 3945 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ficfi 9401 +∞cpnf 11241 -∞cmnf 11242 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 (,)cioo 13320 (,]cioc 13321 [,)cico 13322 topGenctg 17379 ordTopcordt 17441 TosetRel ctsr 18514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-topgen 17385 df-ordt 17443 df-ps 18515 df-tsr 18516 df-top 22387 df-bases 22440 |
This theorem is referenced by: iocpnfordt 22710 icomnfordt 22711 iooordt 22712 pnfnei 22715 mnfnei 22716 xrtgioo 24313 |
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