trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 16184

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 16182 (which means the sequence contains a zero), trilpolemeq1 16181 (which means the sequence is all ones), and trilpolemgt1 16180 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16170) or that the real numbers are a discrete field (see trirec0 16185).

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 10420 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 6780	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 5975	. . . . . . . 8
4	3	oveq2d 5983	. . . . . . 7
5		fveq2 5599	. . . . . . 7
6	4, 5	oveq12d 5985	. . . . . 6
7	6	cbvsumv 11787	. . . . 5
8	2, 7	trilpolemcl 16178	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8122	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4062	. . . . . . . 8
12		eqeq1 2214	. . . . . . . 8
13		breq2 4063	. . . . . . . 8
14	11, 12, 13	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4063	. . . . . . . 8
16		eqeq2 2217	. . . . . . . 8
17		breq1 4062	. . . . . . . 8
18	15, 16, 17	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2898	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1249	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 16183	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2581	. . 3 $\/$ $\/$ $\/$
23		nnex 9077	. . . 4
24		isomninn 16172	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10616	. . 3
28		enomni 7267	. . 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 710 $\/$ w3o 980

wceq 1373

wcel 2178

wral 2486

wrex 2487

cvv 2776

cpr 3644 class class class wbr 4059

com 4656

wf 5286

cfv 5290 (class class class)co 5967

cmap 6758

cen 6848 Omnicomni 7262

cr 7959

cc0 7960

c1 7961

cmul 7965

clt 8142

cdiv 8780

cn 9071

c2 9122

cexp 10720

csu 11779

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-2o 6526 df-oadd 6529 df-er 6643 df-map 6760 df-en 6851 df-dom 6852 df-fin 6853 df-omni 7263 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-ico 10051 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-ihash 10958 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-sumdc 11780

This theorem is referenced by: (None)

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