Theorem dceqnconst 13641

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 13637 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 7866	. . . 4 ⊢ ℝ ∈ V
2	1	mptex 5693	. . 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 9179	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℤ)
5		1zzd 9194	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
6		eqeq1 2164	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
7	6	dcbid 824	. . . . . 6 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 = 0 ↔ DECID 𝑦 = 0))
8	7	rspccva 2815	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → DECID 𝑦 = 0)
9	4, 5, 8	ifcldcd 3540	. . . 4 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → if(𝑦 = 0, 0, 1) ∈ ℤ)
10	9	fmpttd 5622	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ)
11		0re 7878	. . . . . 6 ⊢ 0 ∈ ℝ
12		0zd 9179	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℤ)
13		1zzd 9194	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℤ)
14		eqid 2157	. . . . . . . . . . 11 ⊢ 0 = 0
15	14	orci 721	. . . . . . . . . 10 ⊢ (0 = 0 ∨ ¬ 0 = 0)
16		df-dc 821	. . . . . . . . . 10 ⊢ (DECID 0 = 0 ↔ (0 = 0 ∨ ¬ 0 = 0))
17	15, 16	mpbir 145	. . . . . . . . 9 ⊢ DECID 0 = 0
18	17	a1i 9	. . . . . . . 8 ⊢ (⊤ → DECID 0 = 0)
19	12, 13, 18	ifcldcd 3540	. . . . . . 7 ⊢ (⊤ → if(0 = 0, 0, 1) ∈ ℤ)
20	19	mptru 1344	. . . . . 6 ⊢ if(0 = 0, 0, 1) ∈ ℤ
21		eqeq1 2164	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑦 = 0 ↔ 0 = 0))
22	21	ifbid 3526	. . . . . . 7 ⊢ (𝑦 = 0 → if(𝑦 = 0, 0, 1) = if(0 = 0, 0, 1))
23		eqid 2157	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))
24	22, 23	fvmptg 5544	. . . . . 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 423	. . . . 5 ⊢ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = if(0 = 0, 0, 1)
26	14	iftruei 3511	. . . . 5 ⊢ if(0 = 0, 0, 1) = 0
27	25, 26	eqtri 2178	. . . 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 8901	. . . . . 6 ⊢ 1 ≠ 0
30		eqeq1 2164	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = 0 ↔ 𝑧 = 0))
31	30	ifbid 3526	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → if(𝑦 = 0, 0, 1) = if(𝑧 = 0, 0, 1))
32		rpre 9567	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
33	32	adantl 275	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 𝑧 ∈ ℝ)
34		0zd 9179	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 0 ∈ ℤ)
35		1zzd 9194	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 1 ∈ ℤ)
36		eqeq1 2164	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
37	36	dcbid 824	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 0 ↔ DECID 𝑧 = 0))
38		simpl 108	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ∀𝑥 ∈ ℝ DECID 𝑥 = 0)
39	37, 38, 33	rspcdva 2821	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → DECID 𝑧 = 0)
40	34, 35, 39	ifcldcd 3540	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) ∈ ℤ)
41	23, 31, 33, 40	fvmptd3 5561	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = if(𝑧 = 0, 0, 1))
42		rpne0 9576	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ≠ 0)
43	42	neneqd 2348	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → ¬ 𝑧 = 0)
44	43	iffalsed 3515	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ+ → if(𝑧 = 0, 0, 1) = 1)
45	44	adantl 275	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) = 1)
46	41, 45	eqtrd 2190	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = 1)
47	46	neeq1d 2345	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ 1 ≠ 0))
48	29, 47	mpbiri 167	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
49	48	ralrimiva 2530	. . . 4 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑧 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
50		fveq2 5468	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
51	50	neeq1d 2345	. . . . 5 ⊢ (𝑧 = 𝑥 → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
52	51	cbvralv 2680	. . . 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 121	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑥 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0)
54	10, 28, 53	3jca 1162	. 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 5302	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓:ℝ⟶ℤ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ))
56		fveq1 5467	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘0) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0))
57	56	eqeq1d 2166	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0))
58		fveq1 5467	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘𝑥) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
59	58	neeq1d 2345	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
60	59	ralbidv 2457	. . 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 1294	. 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 2857	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 103   ∨ wo 698  DECID wdc 820   ∧ w3a 963   = wceq 1335  ⊤wtru 1336  ∃wex 1472   ∈ wcel 2128   ≠ wne 2327  ∀wral 2435  Vcvv 2712  ifcif 3505   ↦ cmpt 4025  ⟶wf 5166  ‘cfv 5170  ℝcr 7731  0cc0 7732  1c1 7733  ℤcz 9167  ℝ⁺crp 9560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-z 9168  df-rp 9561

This theorem is referenced by:  dcapnconstALT  13643