Theorem dceqnconst 15550

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 15545 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 8006	. . . 4 ⊢ ℝ ∈ V
2	1	mptex 5784	. . 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 9329	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℤ)
5		1zzd 9344	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
6		eqeq1 2200	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
7	6	dcbid 839	. . . . . 6 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 = 0 ↔ DECID 𝑦 = 0))
8	7	rspccva 2863	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → DECID 𝑦 = 0)
9	4, 5, 8	ifcldcd 3593	. . . 4 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → if(𝑦 = 0, 0, 1) ∈ ℤ)
10	9	fmpttd 5713	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ)
11		0re 8019	. . . . . 6 ⊢ 0 ∈ ℝ
12		0zd 9329	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℤ)
13		1zzd 9344	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℤ)
14		eqid 2193	. . . . . . . . . . 11 ⊢ 0 = 0
15	14	orci 732	. . . . . . . . . 10 ⊢ (0 = 0 ∨ ¬ 0 = 0)
16		df-dc 836	. . . . . . . . . 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 3593	. . . . . . 7 ⊢ (⊤ → if(0 = 0, 0, 1) ∈ ℤ)
20	19	mptru 1373	. . . . . 6 ⊢ if(0 = 0, 0, 1) ∈ ℤ
21		eqeq1 2200	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑦 = 0 ↔ 0 = 0))
22	21	ifbid 3578	. . . . . . 7 ⊢ (𝑦 = 0 → if(𝑦 = 0, 0, 1) = if(0 = 0, 0, 1))
23		eqid 2193	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))
24	22, 23	fvmptg 5633	. . . . . 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 3563	. . . . 5 ⊢ if(0 = 0, 0, 1) = 0
27	25, 26	eqtri 2214	. . . 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 9050	. . . . . 6 ⊢ 1 ≠ 0
30		eqeq1 2200	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = 0 ↔ 𝑧 = 0))
31	30	ifbid 3578	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → if(𝑦 = 0, 0, 1) = if(𝑧 = 0, 0, 1))
32		rpre 9726	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
33	32	adantl 277	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 𝑧 ∈ ℝ)
34		0zd 9329	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 0 ∈ ℤ)
35		1zzd 9344	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 1 ∈ ℤ)
36		eqeq1 2200	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
37	36	dcbid 839	. . . . . . . . . . 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 2869	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → DECID 𝑧 = 0)
40	34, 35, 39	ifcldcd 3593	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) ∈ ℤ)
41	23, 31, 33, 40	fvmptd3 5651	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = if(𝑧 = 0, 0, 1))
42		rpne0 9735	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ≠ 0)
43	42	neneqd 2385	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → ¬ 𝑧 = 0)
44	43	iffalsed 3567	. . . . . . . . 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 2226	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = 1)
47	46	neeq1d 2382	. . . . . 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 2567	. . . 4 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑧 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
50		fveq2 5554	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
51	50	neeq1d 2382	. . . . 5 ⊢ (𝑧 = 𝑥 → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
52	51	cbvralv 2726	. . . 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 1179	. 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 5386	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓:ℝ⟶ℤ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ))
56		fveq1 5553	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘0) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0))
57	56	eqeq1d 2202	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0))
58		fveq1 5553	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘𝑥) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
59	58	neeq1d 2382	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
60	59	ralbidv 2494	. . 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 1323	. 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 2905	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 709  DECID wdc 835   ∧ w3a 980   = wceq 1364  ⊤wtru 1365  ∃wex 1503   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472  Vcvv 2760  ifcif 3557   ↦ cmpt 4090  ⟶wf 5250  ‘cfv 5254  ℝcr 7871  0cc0 7872  1c1 7873  ℤcz 9317  ℝ⁺crp 9719

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-z 9318  df-rp 9720

This theorem is referenced by:  dcapnconstALT  15552