trilpo - Mathbox for Jim Kingdon

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Theorem trilpo 16019

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 16017 (which means the sequence contains a zero), trilpolemeq1 16016 (which means the sequence is all ones), and trilpolemgt1 16015 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16005) or that the real numbers are a discrete field (see trirec0 16020).

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 10385 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 6759	. . . . . 6
2	1	adantl 277	. . . . 5 $\/$ $\/$ $/\$
3		oveq2 5954	. . . . . . . 8
4	3	oveq2d 5962	. . . . . . 7
5		fveq2 5578	. . . . . . 7
6	4, 5	oveq12d 5964	. . . . . 6
7	6	cbvsumv 11705	. . . . 5
8	2, 7	trilpolemcl 16013	. . . . . 6 $\/$ $\/$ $/\$
9		1red 8089	. . . . . 6 $\/$ $\/$ $/\$
10		simpl 109	. . . . . 6 $\/$ $\/$ $/\$ $\/$ $\/$
11		breq1 4048	. . . . . . . 8
12		eqeq1 2212	. . . . . . . 8
13		breq2 4049	. . . . . . . 8
14	11, 12, 13	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
15		breq2 4049	. . . . . . . 8
16		eqeq2 2215	. . . . . . . 8
17		breq1 4048	. . . . . . . 8
18	15, 16, 17	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
19	14, 18	rspc2va 2891	. . . . . 6 $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
20	8, 9, 10, 19	syl21anc 1249	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
21	2, 7, 20	trilpolemres 16018	. . . 4 $\/$ $\/$ $/\$ $\/$
22	21	ralrimiva 2579	. . 3 $\/$ $\/$ $\/$
23		nnex 9044	. . . 4
24		isomninn 16007	. . . 4 Omni $\/$
25	23, 24	ax-mp 5	. . 3 Omni $\/$
26	22, 25	sylibr 134	. 2 $\/$ $\/$ Omni
27		nnenom 10581	. . 3
28		enomni 7243	. . 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 2176

wral 2484

wrex 2485

cvv 2772

cpr 3634 class class class wbr 4045

com 4639

wf 5268

cfv 5272 (class class class)co 5946

cmap 6737

cen 6827 Omnicomni 7238

cr 7926

cc0 7927

c1 7928

cmul 7932

clt 8109

cdiv 8747

cn 9038

c2 9089

cexp 10685

csu 11697

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-isom 5281 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-irdg 6458 df-frec 6479 df-1o 6504 df-2o 6505 df-oadd 6508 df-er 6622 df-map 6739 df-en 6830 df-dom 6831 df-fin 6832 df-omni 7239 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-ico 10018 df-fz 10133 df-fzo 10267 df-seqfrec 10595 df-exp 10686 df-ihash 10923 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-clim 11623 df-sumdc 11698

This theorem is referenced by: (None)

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