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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 23220 | . 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵))) | 
| 4 | letsr 18638 | . . . 4 ⊢ ≤ ∈ TosetRel | |
| 5 | ledm 18635 | . . . . 5 ⊢ ℝ* = dom ≤ | |
| 6 | 1 | leordtvallem1 23218 | . . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| 7 | 1, 2 | leordtvallem2 23219 | . . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) | 
| 8 | leordtval.3 | . . . . . 6 ⊢ 𝐶 = ran (,) | |
| 9 | df-ioo 13391 | . . . . . . . 8 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)}) | |
| 10 | xrltnle 11328 | . . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎)) | |
| 11 | 10 | adantlr 715 | . . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎)) | 
| 12 | xrltnle 11328 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) | |
| 13 | 12 | ancoms 458 | . . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) | 
| 14 | 13 | adantll 714 | . . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦)) | 
| 15 | 11, 14 | anbi12d 632 | . . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ 𝑦 ∈ ℝ*) → ((𝑎 < 𝑦 ∧ 𝑦 < 𝑏) ↔ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦))) | 
| 16 | 15 | rabbidva 3443 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) → {𝑦 ∈ ℝ* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)} = {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) | 
| 17 | 16 | mpoeq3ia 7511 | . . . . . . . 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 2765 | . . . . . . 7 ⊢ (,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) | 
| 19 | 18 | rneqi 5948 | . . . . . 6 ⊢ ran (,) = ran (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) | 
| 20 | 8, 19 | eqtri 2765 | . . . . 5 ⊢ 𝐶 = ran (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)}) | 
| 21 | 5, 6, 7, 20 | ordtbas2 23199 | . . . 4 ⊢ ( ≤ ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 22 | 4, 21 | ax-mp 5 | . . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶) | 
| 23 | 22 | fveq2i 6909 | . 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| 24 | 3, 23 | eqtri 2765 | 1 ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∪ cun 3949 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ficfi 9450 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 (,]cioc 13388 [,)cico 13389 topGenctg 17482 ordTopcordt 17544 TosetRel ctsr 18610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-bases 22953 | 
| This theorem is referenced by: iocpnfordt 23223 icomnfordt 23224 iooordt 23225 pnfnei 23228 mnfnei 23229 xrtgioo 24828 | 
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