Theorem trilpo 15603

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 15601 (which means the sequence contains a zero), trilpolemeq1 15600 (which means the sequence is all ones), and trilpolemgt1 15599 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15589) or that the real numbers are a discrete field (see trirec0 15604).

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 10313 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

Assertion

Ref	Expression
trilpo	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 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.

Step	Hyp	Ref	Expression
1		elmapi 6726	. . . . . 6 ⊢ (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → 𝑓:ℕ⟶{0, 1})
2	1	adantl 277	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 𝑓:ℕ⟶{0, 1})
3		oveq2 5927	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
4	3	oveq2d 5935	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
5		fveq2 5555	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
6	4, 5	oveq12d 5937	. . . . . 6 ⊢ (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = ((1 / (2↑𝑗)) · (𝑓‘𝑗)))
7	6	cbvsumv 11507	. . . . 5 ⊢ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓‘𝑗))
8	2, 7	trilpolemcl 15597	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ)
9		1red 8036	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 1 ∈ ℝ)
10		simpl 109	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
11		breq1 4033	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦))
12		eqeq1 2200	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
13		breq2 4034	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑦 < 𝑥 ↔ 𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
14	11, 12, 13	3orbi123d 1322	. . . . . . 7 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ∨ 𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)))))
15		breq2 4034	. . . . . . . 8 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 1))
16		eqeq2 2203	. . . . . . . 8 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
17		breq1 4033	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ↔ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
18	15, 16, 17	3orbi123d 1322	. . . . . . 7 ⊢ (𝑦 = 1 → ((Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ∨ 𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))) ↔ (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)))))
19	14, 18	rspc2va 2879	. . . . . 6 ⊢ (((Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
20	8, 9, 10, 19	syl21anc 1248	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 1 ∨ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1 ∨ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
21	2, 7, 20	trilpolemres 15602	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (∃𝑧 ∈ ℕ (𝑓‘𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
22	21	ralrimiva 2567	. . 3 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓‘𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
23		nnex 8990	. . . 4 ⊢ ℕ ∈ V
24		isomninn 15591	. . . 4 ⊢ (ℕ ∈ V → (ℕ ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓‘𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)))
25	23, 24	ax-mp 5	. . 3 ⊢ (ℕ ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)(∃𝑧 ∈ ℕ (𝑓‘𝑧) = 0 ∨ ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
26	22, 25	sylibr 134	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ℕ ∈ Omni)
27		nnenom 10508	. . 3 ⊢ ℕ ≈ ω
28		enomni 7200	. . 3 ⊢ (ℕ ≈ ω → (ℕ ∈ Omni ↔ ω ∈ Omni))
29	27, 28	ax-mp 5	. 2 ⊢ (ℕ ∈ Omni ↔ ω ∈ Omni)
30	26, 29	sylib 122	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ω ∈ Omni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∨ w3o 979   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760  {cpr 3620   class class class wbr 4030  ωcom 4623  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919   ↑_𝑚 cmap 6704   ≈ cen 6794  Omnicomni 7195  ℝcr 7873  0cc0 7874  1c1 7875   · cmul 7879   < clt 8056   / cdiv 8693  ℕcn 8984  2c2 9035  ↑cexp 10612  Σcsu 11499

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-2o 6472  df-oadd 6475  df-er 6589  df-map 6706  df-en 6797  df-dom 6798  df-fin 6799  df-omni 7196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ico 9963  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by: (None)