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Mirrors > Home > MPE Home > Th. List > leordtvallem1 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 21017. (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 12257 | . . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ* | |
3 | sseqin2 3817 | . . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ* ↔ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)) | |
4 | 2, 3 | mpbi 220 | . . . . 5 ⊢ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞) |
5 | simpl 473 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
6 | pnfxr 10092 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
7 | elioc1 12217 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | |
8 | 5, 6, 7 | sylancl 694 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
9 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | pnfge 11964 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → 𝑦 ≤ +∞) | |
11 | 9, 10 | jccir 562 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞)) |
12 | 11 | biantrurd 529 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))) |
13 | 3anan32 1050 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)) | |
14 | 12, 13 | syl6bbr 278 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) |
15 | xrltnle 10105 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥)) | |
16 | 8, 14, 15 | 3bitr2d 296 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥)) |
17 | 16 | rabbi2dva 3821 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
18 | 4, 17 | syl5eqr 2670 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (𝑥(,]+∞) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
19 | 18 | mpteq2ia 4740 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
20 | 19 | rneqi 5352 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
21 | 1, 20 | eqtri 2644 | 1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 (class class class)co 6650 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,]cioc 12176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioc 12180 |
This theorem is referenced by: leordtval2 21016 leordtval 21017 |
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