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 17190

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 17188 (which means the sequence contains a zero), trilpolemeq1 17187 (which means the sequence is all ones), and trilpolemgt1 17186 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 17176) or that the real numbers are a discrete field (see trirec0 17191).

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 10685 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 6944	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 6093	. . . . . . . 8
4	3	oveq2d 6101	. . . . . . 7
5		fveq2 5695	. . . . . . 7
6	4, 5	oveq12d 6103	. . . . . 6
7	6	cbvsumv 12143	. . . . 5
8	2, 7	trilpolemcl 17184	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8341	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4133	. . . . . . . 8
12		eqeq1 2245	. . . . . . . 8
13		breq2 4134	. . . . . . . 8
14	11, 12, 13	3orbi123d 1352	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4134	. . . . . . . 8
16		eqeq2 2248	. . . . . . . 8
17		breq1 4133	. . . . . . . 8
18	15, 16, 17	3orbi123d 1352	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2944	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1277	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 17189	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2623	. . 3 $\/$ $\/$ $\/$
23		nnex 9312	. . . 4
24		isomninn 17178	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10884	. . 3
28		enomni 7479	. . 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

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $\/$ w3o 1008

wceq 1402

wcel 2209

wral 2528

wrex 2529

cvv 2821

cpr 3710 class class class wbr 4130

com 4737

wf 5373

cfv 5377 (class class class)co 6085

cmap 6922

cen 7020 Omnicomni 7474

cr 8178

cc0 8179

c1 8180

cmul 8184

clt 8360

cdiv 9004

cn 9306

c2 9357

cexp 10988

csu 12135

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-omni 7475 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-n0 9568 df-z 9649 df-uz 9931 df-q 10029 df-rp 10065 df-ico 10306 df-fz 10422 df-fzo 10560 df-seqfrec 10898 df-exp 10989 df-ihash 11229 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136

This theorem is used by: (None)

W3C validator