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Theorem leordtvallem2 21816

Description: Lemma for leordtval 21818. (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 12810	. . . . . 6 ⊢ (-∞[,)𝑥) ⊆ ℝ^*
3		sseqin2 4142	. . . . . 6 ⊢ ((-∞[,)𝑥) ⊆ ℝ^* ↔ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥))
4	2, 3	mpbi 233	. . . . 5 ⊢ (ℝ^* ∩ (-∞[,)𝑥)) = (-∞[,)𝑥)
5		mnfxr 10687	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6		simpl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
7		elico1 12769	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
8	5, 6, 7	sylancr 590	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (-∞[,)𝑥) ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
9		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		mnfle 12517	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
11	9, 10	jccir 525	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦))
12	11	biantrurd 536	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑥 ↔ ((𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥)))
13		df-3an 1086	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ^* ∧ -∞ ≤ 𝑦) ∧ 𝑦 < 𝑥))
14	12, 13	syl6bbr 292	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑥 ↔ (𝑦 ∈ ℝ^ ∧ -∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
15		xrltnle 10697	. . . . . . . 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 310	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (-∞[,)𝑥) ↔ ¬ 𝑥 ≤ 𝑦))
18	17	rabbi2dva 4144	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (-∞[,)𝑥)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
19	4, 18	syl5eqr 2847	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (-∞[,)𝑥) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
20	19	mpteq2ia 5121	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
21	20	rneqi 5771	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
22	1, 21	eqtri 2821	1 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 (class class class)co 7135 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 (,]cioc 12727 [,)cico 12728

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732

This theorem is referenced by: leordtval2 21817 leordtval 21818

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