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Theorem leordtvallem1 21964

Description: Lemma for leordtval 21967. (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 12908	. . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
3		sseqin2 4107	. . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ^* ↔ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞))
4	2, 3	mpbi 233	. . . . 5 ⊢ (ℝ^* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)
5		simpl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑥 ∈ ℝ^)
6		pnfxr 10776	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
7		elioc1 12866	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
8	5, 6, 7	sylancl 589	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ^ ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
9		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → 𝑦 ∈ ℝ^)
10		pnfge 12611	. . . . . . . . . 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^) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ^ ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)))
13		3anan32 1098	. . . . . . . 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 10789	. . . . . . 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 4109	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
18	4, 17	eqtr3id 2788	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
19	18	mpteq2ia 5122	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
20	19	rneqi 5781	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
21	1, 20	eqtri 2762	1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {crab 3058 ∩ cin 3843 ⊆ wss 3844 class class class wbr 5031 ↦ cmpt 5111 ran crn 5527 (class class class)co 7173 +∞cpnf 10753 ℝ^*cxr 10755 < clt 10756 ≤ cle 10757 (,]cioc 12825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-ioc 12829

This theorem is referenced by: leordtval2 21966 leordtval 21967

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