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Mirrors > Home > MPE Home > Th. List > leordtvallem2 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 23110. (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 13435 | . . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ* | |
3 | sseqin2 4211 | . . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ* ↔ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)) | |
4 | 2, 3 | mpbi 229 | . . . . 5 ⊢ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥) |
5 | mnfxr 11295 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
6 | simpl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
7 | elico1 13393 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) | |
8 | 5, 6, 7 | sylancr 586 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
9 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | mnfle 13140 | . . . . . . . . . 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*) → (𝑦 < 𝑥 ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥))) |
13 | df-3an 1087 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥)) | |
14 | 12, 13 | bitr4di 289 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
15 | xrltnle 11305 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) | |
16 | 15 | ancoms 458 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) |
17 | 8, 14, 16 | 3bitr2d 307 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ ¬ 𝑥 ≤ 𝑦)) |
18 | 17 | rabbi2dva 4213 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
19 | 4, 18 | eqtr3id 2781 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (-∞[,)𝑥) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
20 | 19 | mpteq2ia 5245 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
21 | 20 | rneqi 5933 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
22 | 1, 21 | eqtri 2755 | 1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {crab 3427 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5142 ↦ cmpt 5225 ran crn 5673 (class class class)co 7414 +∞cpnf 11269 -∞cmnf 11270 ℝ*cxr 11271 < clt 11272 ≤ cle 11273 (,]cioc 13351 [,)cico 13352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-ico 13356 |
This theorem is referenced by: leordtval2 23109 leordtval 23110 |
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