trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 16775

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 16773 (which means the sequence contains a zero), trilpolemeq1 16772 (which means the sequence is all ones), and trilpolemgt1 16771 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16761) or that the real numbers are a discrete field (see trirec0 16776).

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 10563 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 6882	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 6036	. . . . . . . 8
4	3	oveq2d 6044	. . . . . . 7
5		fveq2 5648	. . . . . . 7
6	4, 5	oveq12d 6046	. . . . . 6
7	6	cbvsumv 12001	. . . . 5
8	2, 7	trilpolemcl 16769	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8254	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4096	. . . . . . . 8
12		eqeq1 2238	. . . . . . . 8
13		breq2 4097	. . . . . . . 8
14	11, 12, 13	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4097	. . . . . . . 8
16		eqeq2 2241	. . . . . . . 8
17		breq1 4096	. . . . . . . 8
18	15, 16, 17	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2925	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1273	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 16774	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2606	. . 3 $\/$ $\/$ $\/$
23		nnex 9208	. . . 4
24		isomninn 16763	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10759	. . 3
28		enomni 7398	. . 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 2202

wral 2511

wrex 2512

cvv 2803

cpr 3674 class class class wbr 4093

com 4694

wf 5329

cfv 5333 (class class class)co 6028

cmap 6860

cen 6950 Omnicomni 7393

cr 8091

cc0 8092

c1 8093

cmul 8097

clt 8273

cdiv 8911

cn 9202

c2 9253

cexp 10863

csu 11993

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-map 6862 df-en 6953 df-dom 6954 df-fin 6955 df-omni 7394 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-ico 10190 df-fz 10306 df-fzo 10440 df-seqfrec 10773 df-exp 10864 df-ihash 11101 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-clim 11919 df-sumdc 11994

This theorem is referenced by: (None)

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