trilpo - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > trilpo		Unicode version

Theorem trilpo 14932

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 14930 (which means the sequence contains a zero), trilpolemeq1 14929 (which means the sequence is all ones), and trilpolemgt1 14928 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14918) or that the real numbers are a discrete field (see trirec0 14933).

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 10246 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 6673	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 5886	. . . . . . . 8
4	3	oveq2d 5894	. . . . . . 7
5		fveq2 5517	. . . . . . 7
6	4, 5	oveq12d 5896	. . . . . 6
7	6	cbvsumv 11372	. . . . 5
8	2, 7	trilpolemcl 14926	. . . . . 6 $\/$ $\/$ $/\$
9		1red 7975	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4008	. . . . . . . 8
12		eqeq1 2184	. . . . . . . 8
13		breq2 4009	. . . . . . . 8
14	11, 12, 13	3orbi123d 1311	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4009	. . . . . . . 8
16		eqeq2 2187	. . . . . . . 8
17		breq1 4008	. . . . . . . 8
18	15, 16, 17	3orbi123d 1311	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2857	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1237	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 14931	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2550	. . 3 $\/$ $\/$ $\/$
23		nnex 8928	. . . 4
24		isomninn 14920	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10437	. . 3
28		enomni 7140	. . 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 708 $\/$ w3o 977

wceq 1353

wcel 2148

wral 2455

wrex 2456

cvv 2739

cpr 3595 class class class wbr 4005

com 4591

wf 5214

cfv 5218 (class class class)co 5878

cmap 6651

cen 6741 Omnicomni 7135

cr 7813

cc0 7814

c1 7815

cmul 7819

clt 7995

cdiv 8632

cn 8922

c2 8973

cexp 10522

csu 11364

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-frec 6395 df-1o 6420 df-2o 6421 df-oadd 6424 df-er 6538 df-map 6653 df-en 6744 df-dom 6745 df-fin 6746 df-omni 7136 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-ico 9897 df-fz 10012 df-fzo 10146 df-seqfrec 10449 df-exp 10523 df-ihash 10759 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-clim 11290 df-sumdc 11365

This theorem is referenced by: (None)

W3C validator