trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 16411

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 16409 (which means the sequence contains a zero), trilpolemeq1 16408 (which means the sequence is all ones), and trilpolemgt1 16407 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16397) or that the real numbers are a discrete field (see trirec0 16412).

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 10460 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 6817	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 6009	. . . . . . . 8
4	3	oveq2d 6017	. . . . . . 7
5		fveq2 5627	. . . . . . 7
6	4, 5	oveq12d 6019	. . . . . 6
7	6	cbvsumv 11872	. . . . 5
8	2, 7	trilpolemcl 16405	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8161	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4086	. . . . . . . 8
12		eqeq1 2236	. . . . . . . 8
13		breq2 4087	. . . . . . . 8
14	11, 12, 13	3orbi123d 1345	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4087	. . . . . . . 8
16		eqeq2 2239	. . . . . . . 8
17		breq1 4086	. . . . . . . 8
18	15, 16, 17	3orbi123d 1345	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2921	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1270	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 16410	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2603	. . 3 $\/$ $\/$ $\/$
23		nnex 9116	. . . 4
24		isomninn 16399	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10656	. . 3
28		enomni 7306	. . 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 713 $\/$ w3o 1001

wceq 1395

wcel 2200

wral 2508

wrex 2509

cvv 2799

cpr 3667 class class class wbr 4083

com 4682

wf 5314

cfv 5318 (class class class)co 6001

cmap 6795

cen 6885 Omnicomni 7301

cr 7998

cc0 7999

c1 8000

cmul 8004

clt 8181

cdiv 8819

cn 9110

c2 9161

cexp 10760

csu 11864

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-2o 6563 df-oadd 6566 df-er 6680 df-map 6797 df-en 6888 df-dom 6889 df-fin 6890 df-omni 7302 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-ico 10090 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-exp 10761 df-ihash 10998 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-sumdc 11865

This theorem is referenced by: (None)

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