trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 13239

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 13237 (which means the sequence contains a zero), trilpolemeq1 13236 (which means the sequence is all ones), and trilpolemgt1 13235 (which is not possible). (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 6564	. . . . . 6
2	1	adantl 275	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 5782	. . . . . . . 8
4	3	oveq2d 5790	. . . . . . 7
5		fveq2 5421	. . . . . . 7
6	4, 5	oveq12d 5792	. . . . . 6
7	6	cbvsumv 11133	. . . . 5
8	2, 7	trilpolemcl 13233	. . . . . 6 $\/$ $\/$ $/\$
9		1red 7784	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 108	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 3932	. . . . . . . 8
12		eqeq1 2146	. . . . . . . 8
13		breq2 3933	. . . . . . . 8
14	11, 12, 13	3orbi123d 1289	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 3933	. . . . . . . 8
16		eqeq2 2149	. . . . . . . 8
17		breq1 3932	. . . . . . . 8
18	15, 16, 17	3orbi123d 1289	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2803	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1215	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 13238	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2505	. . 3 $\/$ $\/$ $\/$
23		nnex 8729	. . . 4
24		isomninn 13229	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 133	. 2 $\/$ $\/$ Omni
27		nnenom 10210	. . 3
28		enomni 7011	. . 3 Omni Omni
29	27, 28	ax-mp 5	. 2 Omni Omni
30	26, 29	sylib 121	1 $\/$ $\/$ Omni

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697 $\/$ w3o 961

wceq 1331

wcel 1480

wral 2416

wrex 2417

cvv 2686

cpr 3528 class class class wbr 3929

com 4504

wf 5119

cfv 5123 (class class class)co 5774

cmap 6542

cen 6632 Omnicomni 7004

cr 7622

cc0 7623

c1 7624

cmul 7628

clt 7803

cdiv 8435

cn 8723

c2 8774

cexp 10295

csu 11125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-2o 6314 df-oadd 6317 df-er 6429 df-map 6544 df-en 6635 df-dom 6636 df-fin 6637 df-omni 7006 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-ico 9680 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-ihash 10525 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126

This theorem is referenced by: (None)

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