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Theorem leordtvallem1 23528

Description: Lemma for leordtval 23531. (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 13562	. . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ^*
3		sseqin2 4120	. . . . . 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 11363	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
7		elioc1 13518	. . . . . . . 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 13259	. . . . . . . . . 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 11376	. . . . . . 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 4122	. . . . 5 ⊢ (𝑥 ∈ ℝ^* → (ℝ^* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
18	4, 17	eqtr3id 2809	. . . 4 ⊢ (𝑥 ∈ ℝ^* → (𝑥(,]+∞) = {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
19	18	mpteq2ia 5199	. . 3 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
20	19	rneqi 5918	. 2 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
21	1, 20	eqtri 2783	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 1569 ∈ wcel 2144 {crab 3412 ⊆ wss 3895 ∩ cin 4089 class class class wbr 5102 ↦ cmpt 5185 ran crn 5651 (class class class)co 7420 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 (,]cioc 13477

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-ioc 13481

This theorem is used by: leordtval2 23530 leordtval 23531

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