Theorem dceqnconst 16339

Description: Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 16334 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)

Assertion

Ref	Expression
dceqnconst	⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))

Distinct variable group:   𝑥,𝑓

Proof of Theorem dceqnconst

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 8101	. . . 4 ⊢ ℝ ∈ V
2	1	mptex 5838	. . 3 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) ∈ V
3	2	a1i 9	. 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) ∈ V)
4		0zd 9426	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℤ)
5		1zzd 9441	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
6		eqeq1 2216	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
7	6	dcbid 842	. . . . . 6 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 = 0 ↔ DECID 𝑦 = 0))
8	7	rspccva 2886	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → DECID 𝑦 = 0)
9	4, 5, 8	ifcldcd 3620	. . . 4 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → if(𝑦 = 0, 0, 1) ∈ ℤ)
10	9	fmpttd 5763	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ)
11		0re 8114	. . . . . 6 ⊢ 0 ∈ ℝ
12		0zd 9426	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℤ)
13		1zzd 9441	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℤ)
14		eqid 2209	. . . . . . . . . . 11 ⊢ 0 = 0
15	14	orci 735	. . . . . . . . . 10 ⊢ (0 = 0 ∨ ¬ 0 = 0)
16		df-dc 839	. . . . . . . . . 10 ⊢ (DECID 0 = 0 ↔ (0 = 0 ∨ ¬ 0 = 0))
17	15, 16	mpbir 146	. . . . . . . . 9 ⊢ DECID 0 = 0
18	17	a1i 9	. . . . . . . 8 ⊢ (⊤ → DECID 0 = 0)
19	12, 13, 18	ifcldcd 3620	. . . . . . 7 ⊢ (⊤ → if(0 = 0, 0, 1) ∈ ℤ)
20	19	mptru 1384	. . . . . 6 ⊢ if(0 = 0, 0, 1) ∈ ℤ
21		eqeq1 2216	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑦 = 0 ↔ 0 = 0))
22	21	ifbid 3604	. . . . . . 7 ⊢ (𝑦 = 0 → if(𝑦 = 0, 0, 1) = if(0 = 0, 0, 1))
23		eqid 2209	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))
24	22, 23	fvmptg 5683	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ if(0 = 0, 0, 1) ∈ ℤ) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = if(0 = 0, 0, 1))
25	11, 20, 24	mp2an 426	. . . . 5 ⊢ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = if(0 = 0, 0, 1)
26	14	iftruei 3588	. . . . 5 ⊢ if(0 = 0, 0, 1) = 0
27	25, 26	eqtri 2230	. . . 4 ⊢ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0
28	27	a1i 9	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0)
29		1ne0 9146	. . . . . 6 ⊢ 1 ≠ 0
30		eqeq1 2216	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = 0 ↔ 𝑧 = 0))
31	30	ifbid 3604	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → if(𝑦 = 0, 0, 1) = if(𝑧 = 0, 0, 1))
32		rpre 9824	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
33	32	adantl 277	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 𝑧 ∈ ℝ)
34		0zd 9426	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 0 ∈ ℤ)
35		1zzd 9441	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 1 ∈ ℤ)
36		eqeq1 2216	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
37	36	dcbid 842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 0 ↔ DECID 𝑧 = 0))
38		simpl 109	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ∀𝑥 ∈ ℝ DECID 𝑥 = 0)
39	37, 38, 33	rspcdva 2892	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → DECID 𝑧 = 0)
40	34, 35, 39	ifcldcd 3620	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) ∈ ℤ)
41	23, 31, 33, 40	fvmptd3 5701	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = if(𝑧 = 0, 0, 1))
42		rpne0 9833	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ≠ 0)
43	42	neneqd 2401	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → ¬ 𝑧 = 0)
44	43	iffalsed 3592	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → if(𝑧 = 0, 0, 1) = 1)
45	44	adantl 277	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) = 1)
46	41, 45	eqtrd 2242	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = 1)
47	46	neeq1d 2398	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ 1 ≠ 0))
48	29, 47	mpbiri 168	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
49	48	ralrimiva 2583	. . . 4 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑧 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
50		fveq2 5603	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
51	50	neeq1d 2398	. . . . 5 ⊢ (𝑧 = 𝑥 → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
52	51	cbvralv 2745	. . . 4 ⊢ (∀𝑧 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0)
53	49, 52	sylib 122	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0)
54	10, 28, 53	3jca 1182	. 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
55		feq1 5432	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓:ℝ⟶ℤ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ))
56		fveq1 5602	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘0) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0))
57	56	eqeq1d 2218	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0))
58		fveq1 5602	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘𝑥) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
59	58	neeq1d 2398	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
60	59	ralbidv 2510	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0 ↔ ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
61	55, 57, 60	3anbi123d 1327	. 2 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0) ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0)))
62	3, 54, 61	elabd 2928	1 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 712  DECID wdc 838   ∧ w3a 983   = wceq 1375  ⊤wtru 1376  ∃wex 1518   ∈ wcel 2180   ≠ wne 2380  ∀wral 2488  Vcvv 2779  ifcif 3582   ↦ cmpt 4124  ⟶wf 5290  ‘cfv 5294  ℝcr 7966  0cc0 7967  1c1 7968  ℤcz 9414  ℝ⁺crp 9817

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-z 9415  df-rp 9818

This theorem is referenced by:  dcapnconstALT  16341