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Theorem leordtvallem2 23220

Description: Lemma for leordtval 23222. (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 13473	. . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
3		sseqin2 4222	. . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ^* ↔ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥))
4	2, 3	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)
5		mnfxr 11319	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
7		elico1 13431	. . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		mnfle 13178	. . . . . . . . . 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 1088	. . . . . . . 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 11329	. . . . . . . 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 4225	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
19	4, 18	eqtr3id 2790	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
20	19	mpteq2ia 5244	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
21	20	rneqi 5947	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
22	1, 21	eqtri 2764	1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 (class class class)co 7432 +∞cpnf 11293 -∞cmnf 11294 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 (,]cioc 13389 [,)cico 13390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ico 13394

This theorem is referenced by: leordtval2 23221 leordtval 23222

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