iocpnfordt - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iocpnfordt		Structured version Visualization version GIF version

Theorem iocpnfordt 23041

Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
iocpnfordt	⊢ (𝐴(,]+∞) ∈ (ordTop‘ ≤ )

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

Step	Hyp	Ref	Expression
1		eqid 2724	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		eqid 2724	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
3		eqid 2724	. . . . . . . . 9 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 23039	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
5		letop 23032	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
6	4, 5	eqeltrri 2822	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
7		tgclb 22795	. . . . . . 7 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases ↔ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top)
8	6, 7	mpbir 230	. . . . . 6 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases
9		bastg 22791	. . . . . 6 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases → ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))))
10	8, 9	ax-mp 5	. . . . 5 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
11	10, 4	sseqtrri 4011	. . . 4 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (ordTop‘ ≤ )
12		ssun1 4164	. . . . 5 ⊢ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ⊆ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))
13		ssun1 4164	. . . . . 6 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ⊆ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)))
14		eqid 2724	. . . . . . . 8 ⊢ (𝐴(,]+∞) = (𝐴(,]+∞)
15		oveq1 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥(,]+∞) = (𝐴(,]+∞))
16	15	rspceeqv 3625	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴(,]+∞) = (𝐴(,]+∞)) → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
17	14, 16	mpan2 688	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
18		eqid 2724	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
19		ovex 7434	. . . . . . . 8 ⊢ (𝑥(,]+∞) ∈ V
20	18, 19	elrnmpti 5949	. . . . . . 7 ⊢ ((𝐴(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
21	17, 20	sylibr 233	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)))
22	13, 21	sselid 3972	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
23	12, 22	sselid 3972	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
24	11, 23	sselid 3972	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
25	24	adantr 480	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
26		df-ioc 13326	. . . . . 6 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
27	26	ixxf 13331	. . . . 5 ⊢ (,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
28	27	fdmi 6719	. . . 4 ⊢ dom (,] = (ℝ^* × ℝ^*)
29	28	ndmov 7584	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) = ∅)
30		0opn 22728	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (ordTop‘ ≤ ))
31	5, 30	ax-mp 5	. . 3 ⊢ ∅ ∈ (ordTop‘ ≤ )
32	29, 31	eqeltrdi 2833	. 2 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
33	25, 32	pm2.61i 182	1 ⊢ (𝐴(,]+∞) ∈ (ordTop‘ ≤ )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ∪ cun 3938 ⊆ wss 3940 ∅c0 4314 𝒫 cpw 4594 ↦ cmpt 5221 × cxp 5664 ran crn 5667 ‘cfv 6533 (class class class)co 7401 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 (,)cioo 13321 (,]cioc 13322 [,)cico 13323 topGenctg 17382 ordTopcordt 17444 Topctop 22717 TopBasesctb 22770

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-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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-reu 3369 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-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 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-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 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-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-topgen 17388 df-ordt 17446 df-ps 18521 df-tsr 18522 df-top 22718 df-topon 22735 df-bases 22771

This theorem is referenced by: xrlimcnp 26816 pnfneige0 33420 lmxrge0 33421 xlimpnfvlem1 45037

W3C validator