Theorem trilpo 17299

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 17297 (which means the sequence contains a zero), trilpolemeq1 17296 (which means the sequence is all ones), and trilpolemgt1 17295 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 17284) or that the real numbers are a discrete field (see trirec0 17300).

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 10692 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 6948	. . . . . 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 6096	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
4	3	oveq2d 6104	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
5		fveq2 5698	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
6	4, 5	oveq12d 6106	. . . . . 6 ⊢ (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = ((1 / (2↑𝑗)) · (𝑓‘𝑗)))
7	6	cbvsumv 12152	. . . . 5 ⊢ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓‘𝑗))
8	2, 7	trilpolemcl 17293	. . . . . 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 8347	. . . . . 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 4134	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦))
12		eqeq1 2245	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
13		breq2 4135	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑦 < 𝑥 ↔ 𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
14	11, 12, 13	3orbi123d 1352	. . . . . . 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 4135	. . . . . . . 8 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) < 1))
16		eqeq2 2248	. . . . . . . 8 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
17		breq1 4134	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝑦 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ↔ 1 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖))))
18	15, 16, 17	3orbi123d 1352	. . . . . . 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 2944	. . . . . 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 1277	. . . . 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 17298	. . . 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 2623	. . 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 9319	. . . 4 ⊢ ℕ ∈ V
24		isomninn 17286	. . . 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 10892	. . 3 ⊢ ℕ ≈ ω
28		enomni 7485	. . 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821  {cpr 3710   class class class wbr 4131  ωcom 4739  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   ↑_𝑚 cmap 6926   ≈ cen 7024  Omnicomni 7480  ℝcr 8184  0cc0 8185  1c1 8186   · cmul 8190   < clt 8366   / cdiv 9010  ℕcn 9313  2c2 9364  ↑cexp 10996  Σcsu 12144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-en 7027  df-dom 7028  df-fin 7029  df-omni 7481  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-ico 10313  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145

This theorem is used by: (None)