Theorem redcwlpo 16068

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 16067). 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 10394 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 6764	. . . . . . . . 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 5959	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (2↑𝑖) = (2↑𝑗))
5	4	oveq2d 5967	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (1 / (2↑𝑖)) = (1 / (2↑𝑗)))
6		fveq2 5583	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
7	5, 6	oveq12d 5969	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = ((1 / (2↑𝑗)) · (𝑓‘𝑗)))
8	7	cbvsumv 11716	. . . . . . . 8 ⊢ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = Σ𝑗 ∈ ℕ ((1 / (2↑𝑗)) · (𝑓‘𝑗))
9	3, 8	trilpolemcl 16050	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) ∈ ℝ)
10		1red 8094	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑓 ∈ ({0, 1} ↑𝑚 ℕ)) → 1 ∈ ℝ)
11		eqeq1 2213	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (𝑥 = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
12	11	dcbid 840	. . . . . . . 8 ⊢ (𝑥 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) → (DECID 𝑥 = 𝑦 ↔ DECID Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦))
13		eqeq2 2216	. . . . . . . . 9 ⊢ (𝑦 = 1 → (Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 𝑦 ↔ Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝑓‘𝑖)) = 1))
14	13	dcbid 840	. . . . . . . 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 2891	. . . . . . 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 16067	. . . . . 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 840	. . . . 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 2580	. . 3 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑓 ∈ ({0, 1} ↑𝑚 ℕ)DECID ∀𝑧 ∈ ℕ (𝑓‘𝑧) = 1)
22		nnex 9049	. . . 4 ⊢ ℕ ∈ V
23		iswomninn 16063	. . . 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 10586	. . 3 ⊢ ℕ ≈ ω
27		enwomni 7279	. . 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 836   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  {cpr 3635   class class class wbr 4047  ωcom 4642  ⟶wf 5272  ‘cfv 5276  (class class class)co 5951   ↑_𝑚 cmap 6742   ≈ cen 6832  WOmnicwomni 7272  ℝcr 7931  0cc0 7932  1c1 7933   · cmul 7937   / cdiv 8752  ℕcn 9043  2c2 9094  ↑cexp 10690  Σcsu 11708

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-2o 6510  df-oadd 6513  df-er 6627  df-map 6744  df-en 6835  df-dom 6836  df-fin 6837  df-womni 7273  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-ico 10023  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709

This theorem is referenced by: (None)