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Theorem leordtvallem2 22706

Description: Lemma for leordtval 22708. (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 13405	. . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
3		sseqin2 4214	. . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ^* ↔ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥))
4	2, 3	mpbi 229	. . . . 5 ⊢ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)
5		mnfxr 11267	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6		simpl 483	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
7		elico1 13363	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
8	5, 6, 7	sylancr 587	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
9		simpr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		mnfle 13110	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
11	9, 10	jccir 522	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦))
12	11	biantrurd 533	. . . . . . . 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 288	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
15		xrltnle 11277	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
16	15	ancoms 459	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
17	8, 14, 16	3bitr2d 306	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (-∞[,)𝑥) ↔ ¬ 𝑥 ≤ 𝑦))
18	17	rabbi2dva 4216	. . . . 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 5250	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
21	20	rneqi 5934	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 (class class class)co 7405 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 (,]cioc 13321 [,)cico 13322

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ico 13326

This theorem is referenced by: leordtval2 22707 leordtval 22708

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