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Mirrors > Home > MPE Home > Th. List > leordtvallem1 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 23039. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
leordtval.1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) |
Ref | Expression |
---|---|
leordtvallem1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leordtval.1 | . 2 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) | |
2 | iocssxr 13405 | . . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ* | |
3 | sseqin2 4207 | . . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ* ↔ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)) | |
4 | 2, 3 | mpbi 229 | . . . . 5 ⊢ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞) |
5 | simpl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
6 | pnfxr 11265 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
7 | elioc1 13363 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | |
8 | 5, 6, 7 | sylancl 585 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
9 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | pnfge 13107 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → 𝑦 ≤ +∞) | |
11 | 9, 10 | jccir 521 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞)) |
12 | 11 | biantrurd 532 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))) |
13 | 3anan32 1094 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)) | |
14 | 12, 13 | bitr4di 289 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
15 | xrltnle 11278 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥)) | |
16 | 8, 14, 15 | 3bitr2d 307 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥)) |
17 | 16 | rabbi2dva 4209 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
18 | 4, 17 | eqtr3id 2778 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (𝑥(,]+∞) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
19 | 18 | mpteq2ia 5241 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
20 | 19 | rneqi 5926 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
21 | 1, 20 | eqtri 2752 | 1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3424 ∩ cin 3939 ⊆ wss 3940 class class class wbr 5138 ↦ cmpt 5221 ran crn 5667 (class class class)co 7401 +∞cpnf 11242 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 (,]cioc 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-ioc 13326 |
This theorem is referenced by: leordtval2 23038 leordtval 23039 |
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