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Theorem leordtvallem1 23153

Description: Lemma for leordtval 23156. (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 13348	. . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
3		sseqin2 4164	. . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ^* ↔ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞))
4	2, 3	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)
5		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
6		pnfxr 11187	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
7		elioc1 13304	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
8	5, 6, 7	sylancl 587	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
9		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		pnfge 13045	. . . . . . . . . 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^) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)))
13		3anan32 1097	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ^* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))
14	12, 13	bitr4di 289	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
15		xrltnle 11200	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
16	8, 14, 15	3bitr2d 307	. . . . . 6 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥))
17	16	rabbi2dva 4167	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
18	4, 17	eqtr3id 2786	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
19	18	mpteq2ia 5181	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
20	19	rneqi 5884	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
21	1, 20	eqtri 2760	1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3390 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ran crn 5623 (class class class)co 7358 +∞cpnf 11164 ℝ^*cxr 11166 < clt 11167 ≤ cle 11168 (,]cioc 13263

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-ioc 13267

This theorem is referenced by: leordtval2 23155 leordtval 23156

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