hoissrrn2 - Mathbox for Glauco Siliprandi

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Theorem hoissrrn2 46563

Description: A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoissrrn2.kph	⊢ Ⅎ𝑘𝜑
hoissrrn2.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
hoissrrn2.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)

**Assertion**
Ref	Expression
hoissrrn2	⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑_m 𝑋))

Distinct variable group: 𝑘,𝑋 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘)

**Proof of Theorem hoissrrn2**
Step	Hyp	Ref	Expression
1		ovex 7382	. . . . 5 ⊢ (𝐴[,)𝐵) ∈ V
2	1	rgenw 3048	. . . 4 ⊢ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ V
3		ixpssmapg 8855	. . . 4 ⊢ (∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ V → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑_m 𝑋))
4	2, 3	ax-mp 5	. . 3 ⊢ X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑_m 𝑋)
5	4	a1i 11	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑_m 𝑋))
6		reex 11100	. . . 4 ⊢ ℝ ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ V)
8		hoissrrn2.kph	. . . . 5 ⊢ Ⅎ𝑘𝜑
9		hoissrrn2.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
10		hoissrrn2.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
11		icossre 13331	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ⊆ ℝ)
12	9, 10, 11	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴[,)𝐵) ⊆ ℝ)
13	12	ex 412	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → (𝐴[,)𝐵) ⊆ ℝ))
14	8, 13	ralrimi 3227	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ)
15		iunss 4994	. . . 4 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ)
16	14, 15	sylibr 234	. . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ)
17		mapss 8816	. . 3 ⊢ ((ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑_m 𝑋) ⊆ (ℝ ↑_m 𝑋))
18	7, 16, 17	syl2anc 584	. 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑_m 𝑋) ⊆ (ℝ ↑_m 𝑋))
19	5, 18	sstrd 3946	1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑_m 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2109 ∀wral 3044 Vcvv 3436 ⊆ wss 3903 ∪ ciun 4941 (class class class)co 7349 ↑_m cmap 8753 Xcixp 8824 ℝcr 11008 ℝ^*cxr 11148 [,)cico 13250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-ico 13254

This theorem is referenced by: ovnhoilem1 46586 ovnhoilem2 46587 ovnhoi 46588 hoiqssbllem2 46608

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