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 15737

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 15735 (which means the sequence contains a zero), trilpolemeq1 15734 (which means the sequence is all ones), and trilpolemgt1 15733 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15723) or that the real numbers are a discrete field (see trirec0 15738).

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 10335 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 6733	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 5933	. . . . . . . 8
4	3	oveq2d 5941	. . . . . . 7
5		fveq2 5561	. . . . . . 7
6	4, 5	oveq12d 5943	. . . . . 6
7	6	cbvsumv 11531	. . . . 5
8	2, 7	trilpolemcl 15731	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8046	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4037	. . . . . . . 8
12		eqeq1 2203	. . . . . . . 8
13		breq2 4038	. . . . . . . 8
14	11, 12, 13	3orbi123d 1322	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4038	. . . . . . . 8
16		eqeq2 2206	. . . . . . . 8
17		breq1 4037	. . . . . . . 8
18	15, 16, 17	3orbi123d 1322	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2882	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1248	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 15736	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2570	. . 3 $\/$ $\/$ $\/$
23		nnex 9001	. . . 4
24		isomninn 15725	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10531	. . 3
28		enomni 7209	. . 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 709 $\/$ w3o 979

wceq 1364

wcel 2167

wral 2475

wrex 2476

cvv 2763

cpr 3624 class class class wbr 4034

com 4627

wf 5255

cfv 5259 (class class class)co 5925

cmap 6711

cen 6801 Omnicomni 7204

cr 7883

cc0 7884

c1 7885

cmul 7889

clt 8066

cdiv 8704

cn 8995

c2 9046

cexp 10635

csu 11523

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7975 ax-resscn 7976 ax-1cn 7977 ax-1re 7978 ax-icn 7979 ax-addcl 7980 ax-addrcl 7981 ax-mulcl 7982 ax-mulrcl 7983 ax-addcom 7984 ax-mulcom 7985 ax-addass 7986 ax-mulass 7987 ax-distr 7988 ax-i2m1 7989 ax-0lt1 7990 ax-1rid 7991 ax-0id 7992 ax-rnegex 7993 ax-precex 7994 ax-cnre 7995 ax-pre-ltirr 7996 ax-pre-ltwlin 7997 ax-pre-lttrn 7998 ax-pre-apti 7999 ax-pre-ltadd 8000 ax-pre-mulgt0 8001 ax-pre-mulext 8002 ax-arch 8003 ax-caucvg 8004

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6202 df-2nd 6203 df-recs 6367 df-irdg 6432 df-frec 6453 df-1o 6478 df-2o 6479 df-oadd 6482 df-er 6596 df-map 6713 df-en 6804 df-dom 6805 df-fin 6806 df-omni 7205 df-pnf 8068 df-mnf 8069 df-xr 8070 df-ltxr 8071 df-le 8072 df-sub 8204 df-neg 8205 df-reap 8607 df-ap 8614 df-div 8705 df-inn 8996 df-2 9054 df-3 9055 df-4 9056 df-n0 9255 df-z 9332 df-uz 9607 df-q 9699 df-rp 9734 df-ico 9974 df-fz 10089 df-fzo 10223 df-seqfrec 10545 df-exp 10636 df-ihash 10873 df-cj 11012 df-re 11013 df-im 11014 df-rsqrt 11168 df-abs 11169 df-clim 11449 df-sumdc 11524

This theorem is referenced by: (None)

W3C validator