icopnfcld - Metamath Proof Explorer

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Theorem icopnfcld 24639

Description: Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)

**Assertion**
Ref	Expression
icopnfcld	⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))

Proof of Theorem icopnfcld Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnfxr 11275	. . . . . 6 ⊢ -∞ ∈ ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ ∈ ℝ^*)
3		rexr 11264	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
4		pnfxr 11272	. . . . . 6 ⊢ +∞ ∈ ℝ^*
5	4	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ^*)
6		mnflt 13109	. . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
7		ltpnf 13106	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
8		df-ioo 13334	. . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
9		df-ico 13336	. . . . . 6 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
10		xrlenlt 11283	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐴))
11		xrlttr 13125	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐴 ∧ 𝐴 < +∞) → 𝑤 < +∞))
12		xrltletr 13142	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐴 ∧ 𝐴 ≤ 𝑤) → -∞ < 𝑤))
13	8, 9, 10, 8, 11, 12	ixxun 13346	. . . . 5 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < +∞)) → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞))
14	2, 3, 5, 6, 7, 13	syl32anc 1375	. . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = (-∞(,)+∞))
15		ioomax 13405	. . . 4 ⊢ (-∞(,)+∞) = ℝ
16	14, 15	eqtrdi 2782	. . 3 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ)
17		ioossre 13391	. . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ
18	8, 9, 10	ixxdisj 13345	. . . . 5 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅)
19	1, 3, 5, 18	mp3an2i 1462	. . . 4 ⊢ (𝐴 ∈ ℝ → ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅)
20		uneqdifeq 4487	. . . 4 ⊢ (((-∞(,)𝐴) ⊆ ℝ ∧ ((-∞(,)𝐴) ∩ (𝐴[,)+∞)) = ∅) → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞)))
21	17, 19, 20	sylancr 586	. . 3 ⊢ (𝐴 ∈ ℝ → (((-∞(,)𝐴) ∪ (𝐴[,)+∞)) = ℝ ↔ (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞)))
22	16, 21	mpbid 231	. 2 ⊢ (𝐴 ∈ ℝ → (ℝ ∖ (-∞(,)𝐴)) = (𝐴[,)+∞))
23		retop 24633	. . 3 ⊢ (topGen‘ran (,)) ∈ Top
24		iooretop 24637	. . 3 ⊢ (-∞(,)𝐴) ∈ (topGen‘ran (,))
25		uniretop 24634	. . . 4 ⊢ ℝ = ∪ (topGen‘ran (,))
26	25	opncld 22892	. . 3 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (-∞(,)𝐴) ∈ (topGen‘ran (,))) → (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,))))
27	23, 24, 26	mp2an 689	. 2 ⊢ (ℝ ∖ (-∞(,)𝐴)) ∈ (Clsd‘(topGen‘ran (,)))
28	22, 27	eqeltrrdi 2836	1 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ran crn 5670 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 (,)cioo 13330 [,)cico 13332 topGenctg 17392 Topctop 22750 Clsdccld 22875

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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-ioo 13334 df-ico 13336 df-topgen 17398 df-top 22751 df-bases 22804 df-cld 22878

This theorem is referenced by: sxbrsigalem3 33801 orvcgteel 33996 dvasin 37085 dvacos 37086 dvreasin 37087 dvreacos 37088 rfcnpre3 44293

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