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Mirrors > Home > MPE Home > Th. List > leordtvallem1 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 21749. (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 12808 | . . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ* | |
3 | sseqin2 4189 | . . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ* ↔ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)) | |
4 | 2, 3 | mpbi 231 | . . . . 5 ⊢ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞) |
5 | simpl 483 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
6 | pnfxr 10683 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
7 | elioc1 12768 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | |
8 | 5, 6, 7 | sylancl 586 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
9 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | pnfge 12513 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → 𝑦 ≤ +∞) | |
11 | 9, 10 | jccir 522 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞)) |
12 | 11 | biantrurd 533 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))) |
13 | 3anan32 1089 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)) | |
14 | 12, 13 | syl6bbr 290 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
15 | xrltnle 10696 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥)) | |
16 | 8, 14, 15 | 3bitr2d 308 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥)) |
17 | 16 | rabbi2dva 4191 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
18 | 4, 17 | syl5eqr 2867 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (𝑥(,]+∞) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
19 | 18 | mpteq2ia 5148 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
20 | 19 | rneqi 5800 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
21 | 1, 20 | eqtri 2841 | 1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ran crn 5549 (class class class)co 7145 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 (,]cioc 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioc 12731 |
This theorem is referenced by: leordtval2 21748 leordtval 21749 |
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