leordtval - Metamath Proof Explorer

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

Theorem leordtval 22646

Description: The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypotheses**
Ref	Expression
leordtval.1	⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
leordtval.2	⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
leordtval.3	⊢ 𝐶 = ran (,)

**Assertion**
Ref	Expression
leordtval	⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))

Proof of Theorem leordtval Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		leordtval.1	. . 3 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
2		leordtval.2	. . 3 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
3	1, 2	leordtval2 22645	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))
4		letsr 18528	. . . 4 ⊢ ≤ ∈ TosetRel
5		ledm 18525	. . . . 5 ⊢ ℝ^* = dom ≤
6	1	leordtvallem1 22643	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
7	1, 2	leordtvallem2 22644	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
8		leordtval.3	. . . . . 6 ⊢ 𝐶 = ran (,)
9		df-ioo 13310	. . . . . . . 8 ⊢ (,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)})
10		xrltnle 11263	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎))
11	10	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎))
12		xrltnle 11263	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
13	12	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
14	13	adantll 712	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
15	11, 14	anbi12d 631	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → ((𝑎 < 𝑦 ∧ 𝑦 < 𝑏) ↔ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)))
16	15	rabbidva 3438	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → {𝑦 ∈ ℝ^ ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)} = {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
17	16	mpoeq3ia 7471	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)}) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
18	9, 17	eqtri 2759	. . . . . . 7 ⊢ (,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
19	18	rneqi 5928	. . . . . 6 ⊢ ran (,) = ran (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
20	8, 19	eqtri 2759	. . . . 5 ⊢ 𝐶 = ran (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
21	5, 6, 7, 20	ordtbas2 22624	. . . 4 ⊢ ( ≤ ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
22	4, 21	ax-mp 5	. . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)
23	22	fveq2i 6881	. 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))
24	3, 23	eqtri 2759	1 ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3431 ∪ cun 3942 class class class wbr 5141 ↦ cmpt 5224 ran crn 5670 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 ficfi 9387 +∞cpnf 11227 -∞cmnf 11228 ℝ^*cxr 11229 < clt 11230 ≤ cle 11231 (,)cioo 13306 (,]cioc 13307 [,)cico 13308 topGenctg 17365 ordTopcordt 17427 TosetRel ctsr 18500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 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-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fi 9388 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-topgen 17371 df-ordt 17429 df-ps 18501 df-tsr 18502 df-top 22325 df-bases 22378

This theorem is referenced by: iocpnfordt 22648 icomnfordt 22649 iooordt 22650 pnfnei 22653 mnfnei 22654 xrtgioo 24251

W3C validator