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Theorem leordtvallem1 23529

Description: Lemma for leordtval 23532. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypothesis**
Ref	Expression
leordtval.1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))

**Assertion**
Ref	Expression
leordtvallem1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

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

**Proof of Theorem leordtvallem1**
Step	Hyp	Ref	Expression
1		leordtval.1	. 2 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		iocssxr 13563	. . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
3		sseqin2 4169	. . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ^* ↔ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞))
4	2, 3	mpbi 233	. . . . 5 ⊢ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)
5		simpl 488	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
6		pnfxr 11364	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
7		elioc1 13519	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
8	5, 6, 7	sylancl 598	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
9		simpr 490	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		pnfge 13260	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
11	9, 10	jccir 531	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞))
12	11	biantrurd 542	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)))
13		3anan32 1113	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))
14	12, 13	bitr4di 292	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
15		xrltnle 11377	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
16	8, 14, 15	3bitr2d 310	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥))
17	16	rabbi2dva 4171	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
18	4, 17	eqtr3id 2810	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
19	18	mpteq2ia 5200	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
20	19	rneqi 5919	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
21	1, 20	eqtri 2784	1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {crab 3413 ∩ cin 3898 ⊆ wss 3914 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 (class class class)co 7421 +∞cpnf 11341 ℝ^*cxr 11343 < clt 11344 ≤ cle 11345 (,]cioc 13478

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-fv 6547 df-ov 7424 df-oprab 7425 df-mpo 7426 df-1st 8002 df-2nd 8003 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-ioc 13482

This theorem is used by: leordtval2 23531 leordtval 23532

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