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Mirrors > Home > MPE Home > Th. List > leordtvallem2 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 21436. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
leordtval.1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) |
leordtval.2 | ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) |
Ref | Expression |
---|---|
leordtvallem2 | ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leordtval.2 | . 2 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) | |
2 | icossxr 12575 | . . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ* | |
3 | sseqin2 4040 | . . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ* ↔ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)) | |
4 | 2, 3 | mpbi 222 | . . . . 5 ⊢ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥) |
5 | mnfxr 10436 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
6 | simpl 476 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
7 | elico1 12535 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) | |
8 | 5, 6, 7 | sylancr 581 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
9 | simpr 479 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | mnfle 12284 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → -∞ ≤ 𝑦) | |
11 | 9, 10 | jccir 517 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦)) |
12 | 11 | biantrurd 528 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥))) |
13 | df-3an 1073 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥)) | |
14 | 12, 13 | syl6bbr 281 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
15 | xrltnle 10446 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) | |
16 | 15 | ancoms 452 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) |
17 | 8, 14, 16 | 3bitr2d 299 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ ¬ 𝑥 ≤ 𝑦)) |
18 | 17 | rabbi2dva 4042 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
19 | 4, 18 | syl5eqr 2828 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (-∞[,)𝑥) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
20 | 19 | mpteq2ia 4977 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
21 | 20 | rneqi 5599 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
22 | 1, 21 | eqtri 2802 | 1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 {crab 3094 ∩ cin 3791 ⊆ wss 3792 class class class wbr 4888 ↦ cmpt 4967 ran crn 5358 (class class class)co 6924 +∞cpnf 10410 -∞cmnf 10411 ℝ*cxr 10412 < clt 10413 ≤ cle 10414 (,]cioc 12493 [,)cico 12494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-ico 12498 |
This theorem is referenced by: leordtval2 21435 leordtval 21436 |
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