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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoissrrn2 | Structured version Visualization version GIF version |
Description: A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
hoissrrn2.kph | ⊢ Ⅎ𝑘𝜑 |
hoissrrn2.a | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ) |
hoissrrn2.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
hoissrrn2 | ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑𝑚 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6833 | . . . . 5 ⊢ (𝐴[,)𝐵) ∈ V | |
2 | 1 | rgenw 3054 | . . . 4 ⊢ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ V |
3 | ixpssmapg 8096 | . . . 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 10211 | . . . 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 12439 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ) | |
12 | 9, 10, 11 | syl2anc 696 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴[,)𝐵) ⊆ ℝ) |
13 | 12 | ex 449 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → (𝐴[,)𝐵) ⊆ ℝ)) |
14 | 8, 13 | ralrimi 3087 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) |
15 | iunss 4705 | . . . 4 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) | |
16 | 14, 15 | sylibr 224 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) |
17 | mapss 8058 | . . 3 ⊢ ((ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)) | |
18 | 7, 16, 17 | syl2anc 696 | . 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)) |
19 | 5, 18 | sstrd 3746 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑𝑚 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1849 ∈ wcel 2131 ∀wral 3042 Vcvv 3332 ⊆ wss 3707 ∪ ciun 4664 (class class class)co 6805 ↑𝑚 cmap 8015 Xcixp 8066 ℝcr 10119 ℝ*cxr 10257 [,)cico 12362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-pre-lttri 10194 ax-pre-lttrn 10195 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-1st 7325 df-2nd 7326 df-er 7903 df-map 8017 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-ico 12366 |
This theorem is referenced by: ovnhoilem1 41313 ovnhoilem2 41314 ovnhoi 41315 hoiqssbllem2 41335 |
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