Theorem dceqnconst 14463

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 14459 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 7936	. . . 4 ⊢ ℝ ∈ V
2	1	mptex 5738	. . 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 9254	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℤ)
5		1zzd 9269	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
6		eqeq1 2184	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
7	6	dcbid 838	. . . . . 6 ⊢ (𝑥 = 𝑦 → (DECID 𝑥 = 0 ↔ DECID 𝑦 = 0))
8	7	rspccva 2840	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → DECID 𝑦 = 0)
9	4, 5, 8	ifcldcd 3569	. . . 4 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑦 ∈ ℝ) → if(𝑦 = 0, 0, 1) ∈ ℤ)
10	9	fmpttd 5667	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ)
11		0re 7948	. . . . . 6 ⊢ 0 ∈ ℝ
12		0zd 9254	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℤ)
13		1zzd 9269	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℤ)
14		eqid 2177	. . . . . . . . . . 11 ⊢ 0 = 0
15	14	orci 731	. . . . . . . . . 10 ⊢ (0 = 0 ∨ ¬ 0 = 0)
16		df-dc 835	. . . . . . . . . 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 3569	. . . . . . 7 ⊢ (⊤ → if(0 = 0, 0, 1) ∈ ℤ)
20	19	mptru 1362	. . . . . 6 ⊢ if(0 = 0, 0, 1) ∈ ℤ
21		eqeq1 2184	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑦 = 0 ↔ 0 = 0))
22	21	ifbid 3555	. . . . . . 7 ⊢ (𝑦 = 0 → if(𝑦 = 0, 0, 1) = if(0 = 0, 0, 1))
23		eqid 2177	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))
24	22, 23	fvmptg 5588	. . . . . 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 3540	. . . . 5 ⊢ if(0 = 0, 0, 1) = 0
27	25, 26	eqtri 2198	. . . 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 8976	. . . . . 6 ⊢ 1 ≠ 0
30		eqeq1 2184	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = 0 ↔ 𝑧 = 0))
31	30	ifbid 3555	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → if(𝑦 = 0, 0, 1) = if(𝑧 = 0, 0, 1))
32		rpre 9647	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
33	32	adantl 277	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 𝑧 ∈ ℝ)
34		0zd 9254	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 0 ∈ ℤ)
35		1zzd 9269	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → 1 ∈ ℤ)
36		eqeq1 2184	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
37	36	dcbid 838	. . . . . . . . . . 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 2846	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → DECID 𝑧 = 0)
40	34, 35, 39	ifcldcd 3569	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → if(𝑧 = 0, 0, 1) ∈ ℤ)
41	23, 31, 33, 40	fvmptd3 5605	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = if(𝑧 = 0, 0, 1))
42		rpne0 9656	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ≠ 0)
43	42	neneqd 2368	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ+ → ¬ 𝑧 = 0)
44	43	iffalsed 3544	. . . . . . . . 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 2210	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 = 0 ∧ 𝑧 ∈ ℝ+) → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = 1)
47	46	neeq1d 2365	. . . . . 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 2550	. . . 4 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∀𝑧 ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0)
50		fveq2 5511	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
51	50	neeq1d 2365	. . . . 5 ⊢ (𝑧 = 𝑥 → (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑧) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
52	51	cbvralv 2703	. . . 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 1177	. 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 5344	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓:ℝ⟶ℤ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):ℝ⟶ℤ))
56		fveq1 5510	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘0) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0))
57	56	eqeq1d 2186	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘0) = 0))
58		fveq1 5510	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → (𝑓‘𝑥) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥))
59	58	neeq1d 2365	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))‘𝑥) ≠ 0))
60	59	ralbidv 2477	. . 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 1312	. 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 2882	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 708  DECID wdc 834   ∧ w3a 978   = wceq 1353  ⊤wtru 1354  ∃wex 1492   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  Vcvv 2737  ifcif 3534   ↦ cmpt 4061  ⟶wf 5208  ‘cfv 5212  ℝcr 7801  0cc0 7802  1c1 7803  ℤcz 9242  ℝ⁺crp 9640

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-z 9243  df-rp 9641

This theorem is referenced by:  dcapnconstALT  14465