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Theorem leordtval 23038

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 23037	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵)))
4		letsr 18547	. . . 4 ⊢ ≤ ∈ TosetRel
5		ledm 18544	. . . . 5 ⊢ ℝ^* = dom ≤
6	1	leordtvallem1 23035	. . . . 5 ⊢ 𝐴 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑦 ≤ 𝑥})
7	1, 2	leordtvallem2 23036	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ℝ^* ↦ {𝑦 ∈ ℝ^* ∣ ¬ 𝑥 ≤ 𝑦})
8		leordtval.3	. . . . . 6 ⊢ 𝐶 = ran (,)
9		df-ioo 13324	. . . . . . . 8 ⊢ (,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)})
10		xrltnle 11277	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎))
11	10	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (𝑎 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑎))
12		xrltnle 11277	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
13	12	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
14	13	adantll 711	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → (𝑦 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑦))
15	11, 14	anbi12d 630	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ 𝑦 ∈ ℝ^) → ((𝑎 < 𝑦 ∧ 𝑦 < 𝑏) ↔ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)))
16	15	rabbidva 3431	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) → {𝑦 ∈ ℝ^ ∣ (𝑎 < 𝑦 ∧ 𝑦 < 𝑏)} = {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
17	16	mpoeq3ia 7479	. . . . . . . 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 2752	. . . . . . 7 ⊢ (,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
19	18	rneqi 5926	. . . . . 6 ⊢ ran (,) = ran (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
20	8, 19	eqtri 2752	. . . . 5 ⊢ 𝐶 = ran (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑦 ∈ ℝ^* ∣ (¬ 𝑦 ≤ 𝑎 ∧ ¬ 𝑏 ≤ 𝑦)})
21	5, 6, 7, 20	ordtbas2 23016	. . . 4 ⊢ ( ≤ ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))
22	4, 21	ax-mp 5	. . 3 ⊢ (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)
23	22	fveq2i 6884	. 2 ⊢ (topGen‘(fi‘(𝐴 ∪ 𝐵))) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))
24	3, 23	eqtri 2752	1 ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 ∪ cun 3938 class class class wbr 5138 ↦ cmpt 5221 ran crn 5667 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 ficfi 9400 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 (,)cioo 13320 (,]cioc 13321 [,)cico 13322 topGenctg 17381 ordTopcordt 17443 TosetRel ctsr 18519

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 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182

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 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fi 9401 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-topgen 17387 df-ordt 17445 df-ps 18520 df-tsr 18521 df-top 22717 df-bases 22770

This theorem is referenced by: iocpnfordt 23040 icomnfordt 23041 iooordt 23042 pnfnei 23045 mnfnei 23046 xrtgioo 24643

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