![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > snunioo1 | Structured version Visualization version GIF version |
Description: The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
snunioo1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3900 | . 2 ⊢ ((𝐴(,)𝐵) ∪ (𝐴[,]𝐴)) = ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) | |
2 | iccid 12413 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴}) | |
3 | 2 | 3ad2ant1 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐴[,]𝐴) = {𝐴}) |
4 | 3 | uneq2d 3910 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ (𝐴[,]𝐴)) = ((𝐴(,)𝐵) ∪ {𝐴})) |
5 | simp1 1131 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ*) | |
6 | simp2 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ*) | |
7 | xrleid 12176 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
8 | 7 | 3ad2ant1 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐴) |
9 | simp3 1133 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
10 | df-icc 12375 | . . . 4 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
11 | df-ioo 12372 | . . . 4 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
12 | xrltnle 10297 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴)) | |
13 | df-ico 12374 | . . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
14 | xrlelttr 12180 | . . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 < 𝐵) → 𝑤 < 𝐵)) | |
15 | simpl1 1228 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ∈ ℝ*) | |
16 | simpl3 1232 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝑤 ∈ ℝ*) | |
17 | simprr 813 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 < 𝑤) | |
18 | 15, 16, 17 | xrltled 39985 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ≤ 𝑤) |
19 | 18 | ex 449 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤)) |
20 | 10, 11, 12, 13, 14, 19 | ixxun 12384 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
21 | 5, 5, 6, 8, 9, 20 | syl32anc 1485 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵)) |
22 | 1, 4, 21 | 3eqtr3a 2818 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∪ cun 3713 {csn 4321 class class class wbr 4804 (class class class)co 6813 ℝ*cxr 10265 < clt 10266 ≤ cle 10267 (,)cioo 12368 [,)cico 12370 [,]cicc 12371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-pre-lttri 10202 ax-pre-lttrn 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-ioo 12372 df-ico 12374 df-icc 12375 |
This theorem is referenced by: limcresioolb 40378 icocncflimc 40605 volico 40703 fourierdlem48 40874 fouriersw 40951 |
Copyright terms: Public domain | W3C validator |