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Theorem leordtvallem2 23273

Description: Lemma for leordtval 23275. (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 13438	. . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
3		sseqin2 4177	. . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ^* ↔ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥))
4	2, 3	mpbi 232	. . . . 5 ⊢ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)
5		mnfxr 11241	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6		simpl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
7		elico1 13394	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
8	5, 6, 7	sylancr 596	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
9		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		mnfle 13139	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
11	9, 10	jccir 529	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦))
12	11	biantrurd 540	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑥 ↔ ((𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥)))
13		df-3an 1101	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ^* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥))
14	12, 13	bitr4di 291	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
15		xrltnle 11251	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
16	15	ancoms 462	. . . . . . 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 4179	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
19	4, 18	eqtr3id 2813	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
20	19	mpteq2ia 5197	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
21	20	rneqi 5915	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
22	1, 21	eqtri 2787	1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 {crab 3416 ∩ cin 3905 ⊆ wss 3906 class class class wbr 5102 ↦ cmpt 5183 ran crn 5650 (class class class)co 7398 +∞cpnf 11215 -∞cmnf 11216 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 (,]cioc 13352 [,)cico 13353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7401 df-oprab 7402 df-mpo 7403 df-1st 7972 df-2nd 7973 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-ico 13357

This theorem is referenced by: leordtval2 23274 leordtval 23275

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