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Theorem trilpo 13432
 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 13430 (which means the sequence contains a zero), trilpolemeq1 13429 (which means the sequence is all ones), and trilpolemgt1 13428 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 13418) or that the real numbers are a discrete field (see trirec0 13433). 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 10071 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)
Assertion
Ref Expression
trilpo (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ω ∈ Omni)
Distinct variable group:   𝑥,𝑦

Proof of Theorem trilpo
Dummy variables 𝑓 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 6573 . . . . . 6 (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → 𝑓:ℕ⟶{0, 1})
21adantl 275 . . . . 5 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 𝑓:ℕ⟶{0, 1})
3 oveq2 5791 . . . . . . . 8 (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
43oveq2d 5799 . . . . . . 7 (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
5 fveq2 5430 . . . . . . 7 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
64, 5oveq12d 5801 . . . . . 6 (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓𝑖)) = ((1 / (2↑𝑗)) · (𝑓𝑗)))
76cbvsumv 11182 . . . . 5 Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓𝑗))
82, 7trilpolemcl 13426 . . . . . 6 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) ∈ ℝ)
9 1red 7825 . . . . . 6 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 1 ∈ ℝ)
10 simpl 108 . . . . . 6 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
11 breq1 3941 . . . . . . . 8 (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) → (𝑥 < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 𝑦))
12 eqeq1 2147 . . . . . . . 8 (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 𝑦))
13 breq2 3942 . . . . . . . 8 (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) → (𝑦 < 𝑥𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖))))
1411, 12, 133orbi123d 1290 . . . . . . 7 (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) → ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 𝑦 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 𝑦𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)))))
15 breq2 3942 . . . . . . . 8 (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 1))
16 eqeq2 2150 . . . . . . . 8 (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 1))
17 breq1 3941 . . . . . . . 8 (𝑦 = 1 → (𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) ↔ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖))))
1815, 16, 173orbi123d 1290 . . . . . . 7 (𝑦 = 1 → ((Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 𝑦 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 𝑦𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖))) ↔ (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)))))
1914, 18rspc2va 2808 . . . . . 6 (((Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖))))
208, 9, 10, 19syl21anc 1216 . . . . 5 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓𝑖))))
212, 7, 20trilpolemres 13431 . . . 4 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (∃𝑧 ∈ ℕ (𝑓𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓𝑧) = 1))
2221ralrimiva 2509 . . 3 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓𝑧) = 1))
23 nnex 8770 . . . 4 ℕ ∈ V
24 isomninn 13420 . . . 4 (ℕ ∈ V → (ℕ ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓𝑧) = 1)))
2523, 24ax-mp 5 . . 3 (ℕ ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓𝑧) = 1))
2622, 25sylibr 133 . 2 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ℕ ∈ Omni)
27 nnenom 10258 . . 3 ℕ ≈ ω
28 enomni 7021 . . 3 (ℕ ≈ ω → (ℕ ∈ Omni ↔ ω ∈ Omni))
2927, 28ax-mp 5 . 2 (ℕ ∈ Omni ↔ ω ∈ Omni)
3026, 29sylib 121 1 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ω ∈ Omni)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∨ w3o 962   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2690  {cpr 3534   class class class wbr 3938  ωcom 4513  ⟶wf 5128  ‘cfv 5132  (class class class)co 5783   ↑𝑚 cmap 6551   ≈ cen 6641  Omnicomni 7014  ℝcr 7663  0cc0 7664  1c1 7665   · cmul 7669   < clt 7844   / cdiv 8476  ℕcn 8764  2c2 8815  ↑cexp 10343  Σcsu 11174 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511  ax-cnex 7755  ax-resscn 7756  ax-1cn 7757  ax-1re 7758  ax-icn 7759  ax-addcl 7760  ax-addrcl 7761  ax-mulcl 7762  ax-mulrcl 7763  ax-addcom 7764  ax-mulcom 7765  ax-addass 7766  ax-mulass 7767  ax-distr 7768  ax-i2m1 7769  ax-0lt1 7770  ax-1rid 7771  ax-0id 7772  ax-rnegex 7773  ax-precex 7774  ax-cnre 7775  ax-pre-ltirr 7776  ax-pre-ltwlin 7777  ax-pre-lttrn 7778  ax-pre-apti 7779  ax-pre-ltadd 7780  ax-pre-mulgt0 7781  ax-pre-mulext 7782  ax-arch 7783  ax-caucvg 7784 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-po 4227  df-iso 4228  df-iord 4297  df-on 4299  df-ilim 4300  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-isom 5141  df-riota 5739  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-recs 6211  df-irdg 6276  df-frec 6297  df-1o 6322  df-2o 6323  df-oadd 6326  df-er 6438  df-map 6553  df-en 6644  df-dom 6645  df-fin 6646  df-omni 7016  df-pnf 7846  df-mnf 7847  df-xr 7848  df-ltxr 7849  df-le 7850  df-sub 7979  df-neg 7980  df-reap 8381  df-ap 8388  df-div 8477  df-inn 8765  df-2 8823  df-3 8824  df-4 8825  df-n0 9022  df-z 9099  df-uz 9371  df-q 9459  df-rp 9491  df-ico 9727  df-fz 9842  df-fzo 9971  df-seqfrec 10270  df-exp 10344  df-ihash 10574  df-cj 10666  df-re 10667  df-im 10668  df-rsqrt 10822  df-abs 10823  df-clim 11100  df-sumdc 11175 This theorem is referenced by: (None)
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