Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > leordtvallem2 | Structured version Visualization version GIF version |
Description: Lemma for leordtval 21821. (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 12822 | . . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ* | |
3 | sseqin2 4192 | . . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ* ↔ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)) | |
4 | 2, 3 | mpbi 232 | . . . . 5 ⊢ (ℝ* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥) |
5 | mnfxr 10698 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
6 | simpl 485 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
7 | elico1 12782 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) | |
8 | 5, 6, 7 | sylancr 589 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
9 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
10 | mnfle 12530 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → -∞ ≤ 𝑦) | |
11 | 9, 10 | jccir 524 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦)) |
12 | 11 | biantrurd 535 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥))) |
13 | df-3an 1085 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥)) | |
14 | 12, 13 | syl6bbr 291 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥))) |
15 | xrltnle 10708 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) | |
16 | 15 | ancoms 461 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) |
17 | 8, 14, 16 | 3bitr2d 309 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (-∞[,)𝑥) ↔ ¬ 𝑥 ≤ 𝑦)) |
18 | 17 | rabbi2dva 4194 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
19 | 4, 18 | syl5eqr 2870 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (-∞[,)𝑥) = {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
20 | 19 | mpteq2ia 5157 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
21 | 20 | rneqi 5807 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
22 | 1, 21 | eqtri 2844 | 1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {crab 3142 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 (class class class)co 7156 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 (,]cioc 12740 [,)cico 12741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ico 12745 |
This theorem is referenced by: leordtval2 21820 leordtval 21821 |
Copyright terms: Public domain | W3C validator |