iocpnfordt - Metamath Proof Explorer

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Theorem iocpnfordt 23163

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 2725	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		eqid 2725	. . . . . . . . 9 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
3		eqid 2725	. . . . . . . . 9 ⊢ ran (,) = ran (,)
4	1, 2, 3	leordtval 23161	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
5		letop 23154	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
6	4, 5	eqeltrri 2822	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
7		tgclb 22917	. . . . . . 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 22913	. . . . . 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 4014	. . . 4 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ⊆ (ordTop‘ ≤ )
12		ssun1 4170	. . . . 5 ⊢ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ⊆ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))
13		ssun1 4170	. . . . . 6 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ⊆ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)))
14		eqid 2725	. . . . . . . 8 ⊢ (𝐴(,]+∞) = (𝐴(,]+∞)
15		oveq1 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥(,]+∞) = (𝐴(,]+∞))
16	15	rspceeqv 3628	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴(,]+∞) = (𝐴(,]+∞)) → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
17	14, 16	mpan2 689	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → ∃𝑥 ∈ ℝ^* (𝐴(,]+∞) = (𝑥(,]+∞))
18		eqid 2725	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
19		ovex 7452	. . . . . . . 8 ⊢ (𝑥(,]+∞) ∈ V
20	18, 19	elrnmpti 5962	. . . . . . 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 3974	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
23	12, 22	sselid 3974	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
24	11, 23	sselid 3974	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
25	24	adantr 479	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) ∈ (ordTop‘ ≤ ))
26		df-ioc 13364	. . . . . 6 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
27	26	ixxf 13369	. . . . 5 ⊢ (,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
28	27	fdmi 6734	. . . 4 ⊢ dom (,] = (ℝ^* × ℝ^*)
29	28	ndmov 7605	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴(,]+∞) = ∅)
30		0opn 22850	. . . 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 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∪ cun 3942 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 ↦ cmpt 5232 × cxp 5676 ran crn 5679 ‘cfv 6549 (class class class)co 7419 +∞cpnf 11277 -∞cmnf 11278 ℝ^*cxr 11279 < clt 11280 ≤ cle 11281 (,)cioo 13359 (,]cioc 13360 [,)cico 13361 topGenctg 17422 ordTopcordt 17484 Topctop 22839 TopBasesctb 22892

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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-topgen 17428 df-ordt 17486 df-ps 18561 df-tsr 18562 df-top 22840 df-topon 22857 df-bases 22893

This theorem is referenced by: xrlimcnp 26944 pnfneige0 33671 lmxrge0 33672 xlimpnfvlem1 45351

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