Theorem redcwlpo 16771

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 16770). 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 10550 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 109	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
2		elmapi 6882	. . . . . . . . 9 ⊢ (𝑓 ∈ ({0, 1} ↑𝑚 ℕ) → 𝑓:ℕ⟶{0, 1})
3	2	adantl 277	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 𝑓:ℕ⟶{0, 1})
4		oveq2 6036	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
5	4	oveq2d 6044	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
6		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
7	5, 6	oveq12d 6046	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = ((1 / (2↑𝑗)) · (𝑓‘𝑗)))
8	7	cbvsumv 11984	. . . . . . . 8 ⊢ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓‘𝑗))
9	3, 8	trilpolemcl 16752	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ)
10		1red 8237	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 1 ∈ ℝ)
11		eqeq1 2238	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
12	11	dcbid 846	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (DECID 𝑥 = 𝑦 ↔ DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
13		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
14	13	dcbid 846	. . . . . . . 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 2924	. . . . . . 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 411	. . . . . 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 16770	. . . . . 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 846	. . . . 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 147	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
21	20	ralrimiva 2606	. . 3 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
22		nnex 9191	. . . 4 ⊢ ℕ ∈ V
23		iswomninn 16766	. . . 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 134	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ℕ ∈ WOmni)
26		nnenom 10742	. . 3 ⊢ ℕ ≈ ω
27		enwomni 7412	. . 3 ⊢ (ℕ ≈ ω → (ℕ ∈ WOmni ↔ ω ∈ WOmni))
28	26, 27	ax-mp 5	. 2 ⊢ (ℕ ∈ WOmni ↔ ω ∈ WOmni)
29	25, 28	sylib 122	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803  {cpr 3674   class class class wbr 4093  ωcom 4694  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860   ≈ cen 6950  WOmnicwomni 7405  ℝcr 8074  0cc0 8075  1c1 8076   · cmul 8080   / cdiv 8894  ℕcn 9185  2c2 9236  ↑cexp 10846  Σcsu 11976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-womni 7406  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-ico 10173  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977

This theorem is referenced by: (None)