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Theorem trilpo 13585
 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 13583 (which means the sequence contains a zero), trilpolemeq1 13582 (which means the sequence is all ones), and trilpolemgt1 13581 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 13571) or that the real numbers are a discrete field (see trirec0 13586). 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 10135 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.
StepHypRef Expression
1 elmapi 6612 . . . . . 6
21adantl 275 . . . . 5
3 oveq2 5829 . . . . . . . 8
43oveq2d 5837 . . . . . . 7
5 fveq2 5467 . . . . . . 7
64, 5oveq12d 5839 . . . . . 6
76cbvsumv 11251 . . . . 5
82, 7trilpolemcl 13579 . . . . . 6
9 1red 7887 . . . . . 6
10 simpl 108 . . . . . 6
11 breq1 3968 . . . . . . . 8
12 eqeq1 2164 . . . . . . . 8
13 breq2 3969 . . . . . . . 8
1411, 12, 133orbi123d 1293 . . . . . . 7
15 breq2 3969 . . . . . . . 8
16 eqeq2 2167 . . . . . . . 8
17 breq1 3968 . . . . . . . 8
1815, 16, 173orbi123d 1293 . . . . . . 7
1914, 18rspc2va 2830 . . . . . 6
208, 9, 10, 19syl21anc 1219 . . . . 5
212, 7, 20trilpolemres 13584 . . . 4
2221ralrimiva 2530 . . 3
23 nnex 8833 . . . 4
24 isomninn 13573 . . . 4 Omni
2523, 24ax-mp 5 . . 3 Omni
2622, 25sylibr 133 . 2 Omni
27 nnenom 10326 . . 3
28 enomni 7076 . . 3 Omni Omni
2927, 28ax-mp 5 . 2 Omni Omni
3026, 29sylib 121 1 Omni
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wo 698   w3o 962   wceq 1335   wcel 2128  wral 2435  wrex 2436  cvv 2712  cpr 3561   class class class wbr 3965  com 4548  wf 5165  cfv 5169  (class class class)co 5821   cmap 6590   cen 6680  Omnicomni 7071  cr 7725  cc0 7726  c1 7727   cmul 7731   clt 7906   cdiv 8539  cn 8827  c2 8878  cexp 10411  csu 11243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-2o 6361  df-oadd 6364  df-er 6477  df-map 6592  df-en 6683  df-dom 6684  df-fin 6685  df-omni 7072  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-ico 9791  df-fz 9906  df-fzo 10035  df-seqfrec 10338  df-exp 10412  df-ihash 10643  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-clim 11169  df-sumdc 11244 This theorem is referenced by: (None)
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