leordtvallem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leordtvallem2		Structured version Visualization version GIF version

Theorem leordtvallem2 23159

Description: Lemma for leordtval 23161. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypotheses**
Ref	Expression
leordtval.1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
leordtval.2	⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))

**Assertion**
Ref	Expression
leordtvallem2	⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐵(𝑥,𝑦)

**Proof of Theorem leordtvallem2**
Step	Hyp	Ref	Expression
1		leordtval.2	. 2 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
2		icossxr 13352	. . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
3		sseqin2 4176	. . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ^* ↔ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥))
4	2, 3	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)
5		mnfxr 11193	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
7		elico1 13308	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
8	5, 6, 7	sylancr 588	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
9		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		mnfle 13053	. . . . . . . . . 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 1089	. . . . . . . 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 11203	. . . . . . . 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 4179	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
19	4, 18	eqtr3id 2786	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
20	19	mpteq2ia 5194	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
21	20	rneqi 5887	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
22	1, 21	eqtri 2760	1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3400 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5099 ↦ cmpt 5180 ran crn 5626 (class class class)co 7360 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,]cioc 13266 [,)cico 13267

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-ico 13271

This theorem is referenced by: leordtval2 23160 leordtval 23161

W3C validator