Theorem redcwlpo 13934

Description: Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). 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 is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 13933). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO 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 qdceq 10182 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
redcwlpo	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)

Distinct variable group:   𝑥,𝑦

Proof of Theorem redcwlpo

Dummy variables 𝑓 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
2		elmapi 6636	. . . . . . . . 9 ⊢ (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → 𝑓:ℕ⟶{0, 1})
3	2	adantl 275	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 𝑓:ℕ⟶{0, 1})
4		oveq2 5850	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
5	4	oveq2d 5858	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
6		fveq2 5486	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
7	5, 6	oveq12d 5860	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = ((1 / (2↑𝑗)) · (𝑓‘𝑗)))
8	7	cbvsumv 11302	. . . . . . . 8 ⊢ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓‘𝑗))
9	3, 8	trilpolemcl 13916	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ)
10		1red 7914	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 1 ∈ ℝ)
11		eqeq1 2172	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
12	11	dcbid 828	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (DECID 𝑥 = 𝑦 ↔ DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
13		eqeq2 2175	. . . . . . . . 9 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
14	13	dcbid 828	. . . . . . . 8 ⊢ (𝑦 = 1 → (DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
15	12, 14	rspc2v 2843	. . . . . . 7 ⊢ ((Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
16	9, 10, 15	syl2anc 409	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
17	1, 16	mpd 13	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1)
18	3, 8	redcwlpolemeq1 13933	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1 ↔ ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
19	18	dcbid 828	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → (DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1 ↔ DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
20	17, 19	mpbid 146	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
21	20	ralrimiva 2539	. . 3 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
22		nnex 8863	. . . 4 ⊢ ℕ ∈ V
23		iswomninn 13929	. . . 4 ⊢ (ℕ ∈ V → (ℕ ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1))
24	22, 23	ax-mp 5	. . 3 ⊢ (ℕ ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
25	21, 24	sylibr 133	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ℕ ∈ WOmni)
26		nnenom 10369	. . 3 ⊢ ℕ ≈ ω
27		enwomni 7134	. . 3 ⊢ (ℕ ≈ ω → (ℕ ∈ WOmni ↔ ω ∈ WOmni))
28	26, 27	ax-mp 5	. 2 ⊢ (ℕ ∈ WOmni ↔ ω ∈ WOmni)
29	25, 28	sylib 121	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824   = wceq 1343   ∈ wcel 2136  ∀wral 2444  Vcvv 2726  {cpr 3577   class class class wbr 3982  ωcom 4567  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842   ↑_𝑚 cmap 6614   ≈ cen 6704  WOmnicwomni 7127  ℝcr 7752  0cc0 7753  1c1 7754   · cmul 7758   / cdiv 8568  ℕcn 8857  2c2 8908  ↑cexp 10454  Σcsu 11294

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-2o 6385  df-oadd 6388  df-er 6501  df-map 6616  df-en 6707  df-dom 6708  df-fin 6709  df-womni 7128  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by: (None)