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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 7434 | . . . . 5 ⊢ (𝐴[,)𝐵) ∈ V | |
2 | 1 | rgenw 3057 | . . . 4 ⊢ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ V |
3 | ixpssmapg 8917 | . . . 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 11196 | . . . 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 13401 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ) | |
12 | 9, 10, 11 | syl2anc 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴[,)𝐵) ⊆ ℝ) |
13 | 12 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → (𝐴[,)𝐵) ⊆ ℝ)) |
14 | 8, 13 | ralrimi 3246 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) |
15 | iunss 5038 | . . . 4 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) | |
16 | 14, 15 | sylibr 233 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) |
17 | mapss 8878 | . . 3 ⊢ ((ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ ℝ) → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)) | |
18 | 7, 16, 17 | syl2anc 583 | . 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝑋 (𝐴[,)𝐵) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)) |
19 | 5, 18 | sstrd 3984 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑m 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⊆ wss 3940 ∪ ciun 4987 (class class class)co 7401 ↑m cmap 8815 Xcixp 8886 ℝcr 11104 ℝ*cxr 11243 [,)cico 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-pre-lttri 11179 ax-pre-lttrn 11180 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ico 13326 |
This theorem is referenced by: ovnhoilem1 45768 ovnhoilem2 45769 ovnhoi 45770 hoiqssbllem2 45790 |
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