iocpnfordt - Metamath Proof Explorer

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Theorem iocpnfordt 22711

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 2733	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		eqid 2733	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
3		eqid 2733	. . . . . . . . 9 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 22709	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
5		letop 22702	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
6	4, 5	eqeltrri 2831	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
7		tgclb 22465	. . . . . . 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 22461	. . . . . 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 4019	. . . 4 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (ordTop‘ ≤ )
12		ssun1 4172	. . . . 5 ⊢ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ⊆ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))
13		ssun1 4172	. . . . . 6 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ⊆ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)))
14		eqid 2733	. . . . . . . 8 ⊢ (𝐴(,]+∞) = (𝐴(,]+∞)
15		oveq1 7413	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥(,]+∞) = (𝐴(,]+∞))
16	15	rspceeqv 3633	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴(,]+∞) = (𝐴(,]+∞)) → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
17	14, 16	mpan2 690	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
18		eqid 2733	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
19		ovex 7439	. . . . . . . 8 ⊢ (𝑥(,]+∞) ∈ V
20	18, 19	elrnmpti 5958	. . . . . . 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 3980	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
23	12, 22	sselid 3980	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
24	11, 23	sselid 3980	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
25	24	adantr 482	. 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 6727	. . . 4 ⊢ dom (,] = (ℝ^* × ℝ^*)
29	28	ndmov 7588	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) = ∅)
30		0opn 22398	. . . 4 ⊢ ((ordTop‘ ≤ ) ∈ Top → ∅ ∈ (ordTop‘ ≤ ))
31	5, 30	ax-mp 5	. . 3 ⊢ ∅ ∈ (ordTop‘ ≤ )
32	29, 31	eqeltrdi 2842	. 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 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ‘cfv 6541 (class class class)co 7406 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 (,)cioo 13321 (,]cioc 13322 [,)cico 13323 topGenctg 17380 ordTopcordt 17442 Topctop 22387 TopBasesctb 22440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 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 17386 df-ordt 17444 df-ps 18516 df-tsr 18517 df-top 22388 df-topon 22405 df-bases 22441

This theorem is referenced by: xrlimcnp 26463 pnfneige0 32920 lmxrge0 32921 xlimpnfvlem1 44539

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