trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 16968

Description: Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 16966 (which means the sequence contains a zero), trilpolemeq1 16965 (which means the sequence is all ones), and trilpolemgt1 16964 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16954) or that the real numbers are a discrete field (see trirec0 16969).

LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10628 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

**Assertion**
Ref	Expression
trilpo	$\/$ $\/$ Omni

Distinct variable group:

Proof of Theorem trilpo Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 6918	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 6067	. . . . . . . 8
4	3	oveq2d 6075	. . . . . . 7
5		fveq2 5676	. . . . . . 7
6	4, 5	oveq12d 6077	. . . . . 6
7	6	cbvsumv 12076	. . . . 5
8	2, 7	trilpolemcl 16962	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8306	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4118	. . . . . . . 8
12		eqeq1 2241	. . . . . . . 8
13		breq2 4119	. . . . . . . 8
14	11, 12, 13	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4119	. . . . . . . 8
16		eqeq2 2244	. . . . . . . 8
17		breq1 4118	. . . . . . . 8
18	15, 16, 17	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2938	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1273	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 16967	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2617	. . 3 $\/$ $\/$ $\/$
23		nnex 9264	. . . 4
24		isomninn 16956	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10824	. . 3
28		enomni 7444	. . 3 Omni Omni
29	27, 28	ax-mp 5	. 2 Omni Omni
30	26, 29	sylib 122	1 $\/$ $\/$ Omni

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $\/$ w3o 1004

wceq 1398

wcel 2205

wral 2522

wrex 2523

cvv 2815

cpr 3696 class class class wbr 4115

com 4718

wf 5354

cfv 5358 (class class class)co 6059

cmap 6896

cen 6987 Omnicomni 7439

cr 8143

cc0 8144

c1 8145

cmul 8149

clt 8325

cdiv 8967

cn 9258

c2 9309

cexp 10928

csu 12068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4231 ax-sep 4234 ax-nul 4242 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4665 ax-iinf 4716 ax-cnex 8235 ax-resscn 8236 ax-1cn 8237 ax-1re 8238 ax-icn 8239 ax-addcl 8240 ax-addrcl 8241 ax-mulcl 8242 ax-mulrcl 8243 ax-addcom 8244 ax-mulcom 8245 ax-addass 8246 ax-mulass 8247 ax-distr 8248 ax-i2m1 8249 ax-0lt1 8250 ax-1rid 8251 ax-0id 8252 ax-rnegex 8253 ax-precex 8254 ax-cnre 8255 ax-pre-ltirr 8256 ax-pre-ltwlin 8257 ax-pre-lttrn 8258 ax-pre-apti 8259 ax-pre-ltadd 8260 ax-pre-mulgt0 8261 ax-pre-mulext 8262 ax-arch 8263 ax-caucvg 8264

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3626 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-int 3956 df-iun 3999 df-br 4116 df-opab 4178 df-mpt 4179 df-tr 4215 df-id 4420 df-po 4423 df-iso 4424 df-iord 4493 df-on 4495 df-ilim 4496 df-suc 4498 df-iom 4719 df-xp 4761 df-rel 4762 df-cnv 4763 df-co 4764 df-dm 4765 df-rn 4766 df-res 4767 df-ima 4768 df-iota 5318 df-fun 5360 df-fn 5361 df-f 5362 df-f1 5363 df-fo 5364 df-f1o 5365 df-fv 5366 df-isom 5367 df-riota 6012 df-ov 6062 df-oprab 6063 df-mpo 6064 df-1st 6348 df-2nd 6349 df-recs 6550 df-irdg 6615 df-frec 6636 df-1o 6661 df-2o 6662 df-oadd 6665 df-er 6781 df-map 6898 df-en 6990 df-dom 6991 df-fin 6992 df-omni 7440 df-pnf 8327 df-mnf 8328 df-xr 8329 df-ltxr 8330 df-le 8331 df-sub 8464 df-neg 8465 df-reap 8868 df-ap 8875 df-div 8968 df-inn 9259 df-2 9317 df-3 9318 df-4 9319 df-n0 9518 df-z 9599 df-uz 9876 df-q 9974 df-rp 10009 df-ico 10250 df-fz 10366 df-fzo 10503 df-seqfrec 10838 df-exp 10929 df-ihash 11168 df-cj 11556 df-re 11557 df-im 11558 df-rsqrt 11713 df-abs 11714 df-clim 11994 df-sumdc 12069

This theorem is referenced by: (None)

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