dcapnconst - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > dcapnconst		GIF version

Theorem dcapnconst 14778

Description: Decidability of real number apartness 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 trilpo 14761 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 14777 and in fact this theorem can be proved using dceqnconst 14777 as shown at dcapnconstALT 14779. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

**Assertion**
Ref	Expression
dcapnconst	⊢ (∀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 dcapnconst Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 7944	. . . 4 ⊢ ℝ ∈ V
2	1	mptex 5742	. . 3 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) ∈ V
3	2	a1i 9	. 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) ∈ V)
4		1zzd 9279	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℤ)
5		0zd 9264	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℤ)
6		breq1 4006	. . . . . . 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 3570	. . . 4 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑦 ∈ ℝ) → if(𝑦 # 0, 1, 0) ∈ ℤ)
10	9	fmpttd 5671	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)):ℝ⟶ℤ)
11		0re 7956	. . . . . 6 ⊢ 0 ∈ ℝ
12		1zzd 9279	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℤ)
13		0zd 9264	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℤ)
14		0cn 7948	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
15		apirr 8561	. . . . . . . . . . . 12 ⊢ (0 ∈ ℂ → ¬ 0 # 0)
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ ¬ 0 # 0
17	16	olci 732	. . . . . . . . . 10 ⊢ (0 # 0 ∨ ¬ 0 # 0)
18		df-dc 835	. . . . . . . . . 10 ⊢ (DECID 0 # 0 ↔ (0 # 0 ∨ ¬ 0 # 0))
19	17, 18	mpbir 146	. . . . . . . . 9 ⊢ DECID 0 # 0
20	19	a1i 9	. . . . . . . 8 ⊢ (⊤ → DECID 0 # 0)
21	12, 13, 20	ifcldcd 3570	. . . . . . 7 ⊢ (⊤ → if(0 # 0, 1, 0) ∈ ℤ)
22	21	mptru 1362	. . . . . 6 ⊢ if(0 # 0, 1, 0) ∈ ℤ
23		breq1 4006	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑦 # 0 ↔ 0 # 0))
24	23	ifbid 3555	. . . . . . 7 ⊢ (𝑦 = 0 → if(𝑦 # 0, 1, 0) = if(0 # 0, 1, 0))
25		eqid 2177	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))
26	24, 25	fvmptg 5592	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ if(0 # 0, 1, 0) ∈ ℤ) → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = if(0 # 0, 1, 0))
27	11, 22, 26	mp2an 426	. . . . 5 ⊢ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = if(0 # 0, 1, 0)
28	16	iffalsei 3543	. . . . 5 ⊢ if(0 # 0, 1, 0) = 0
29	27, 28	eqtri 2198	. . . 4 ⊢ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = 0
30	29	a1i 9	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = 0)
31		1ne0 8986	. . . . . 6 ⊢ 1 ≠ 0
32		breq1 4006	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 # 0 ↔ 𝑧 # 0))
33	32	ifbid 3555	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → if(𝑦 # 0, 1, 0) = if(𝑧 # 0, 1, 0))
34		rpre 9659	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → 𝑧 ∈ ℝ)
36		1zzd 9279	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → 1 ∈ ℤ)
37		0zd 9264	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → 0 ∈ ℤ)
38		breq1 4006	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 # 0 ↔ 𝑧 # 0))
39	38	dcbid 838	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 # 0 ↔ DECID 𝑧 # 0))
40		simpl 109	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → ∀𝑥 ∈ ℝ DECID 𝑥 # 0)
41	39, 40, 35	rspcdva 2846	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → DECID 𝑧 # 0)
42	36, 37, 41	ifcldcd 3570	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → if(𝑧 # 0, 1, 0) ∈ ℤ)
43	25, 33, 35, 42	fvmptd3 5609	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) = if(𝑧 # 0, 1, 0))
44		rpap0 9669	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 # 0)
45	44	iftrued 3541	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ⁺ → if(𝑧 # 0, 1, 0) = 1)
46	45	adantl 277	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → if(𝑧 # 0, 1, 0) = 1)
47	43, 46	eqtrd 2210	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) = 1)
48	47	neeq1d 2365	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → (((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) ≠ 0 ↔ 1 ≠ 0))
49	31, 48	mpbiri 168	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ DECID 𝑥 # 0 ∧ 𝑧 ∈ ℝ⁺) → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) ≠ 0)
50	49	ralrimiva 2550	. . . 4 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∀𝑧 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) ≠ 0)
51		fveq2 5515	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) = ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥))
52	51	neeq1d 2365	. . . . 5 ⊢ (𝑧 = 𝑥 → (((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0))
53	52	cbvralv 2703	. . . 4 ⊢ (∀𝑧 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑧) ≠ 0 ↔ ∀𝑥 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0)
54	50, 53	sylib 122	. . 3 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∀𝑥 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0)
55	10, 30, 54	3jca 1177	. 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)):ℝ⟶ℤ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = 0 ∧ ∀𝑥 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0))
56		feq1 5348	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → (𝑓:ℝ⟶ℤ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)):ℝ⟶ℤ))
57		fveq1 5514	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → (𝑓‘0) = ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0))
58	57	eqeq1d 2186	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → ((𝑓‘0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = 0))
59		fveq1 5514	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → (𝑓‘𝑥) = ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥))
60	59	neeq1d 2365	. . . 4 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0))
61	60	ralbidv 2477	. . 3 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → (∀𝑥 ∈ ℝ⁺ (𝑓‘𝑥) ≠ 0 ↔ ∀𝑥 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0))
62	56, 58, 61	3anbi123d 1312	. 2 ⊢ (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)) → ((𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ⁺ (𝑓‘𝑥) ≠ 0) ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0)):ℝ⟶ℤ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘0) = 0 ∧ ∀𝑥 ∈ ℝ⁺ ((𝑦 ∈ ℝ ↦ if(𝑦 # 0, 1, 0))‘𝑥) ≠ 0)))
63	3, 55, 62	elabd 2882	1 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ⁺ (𝑓‘𝑥) ≠ 0))

Colors of variables: wff set class

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 class class class wbr 4003 ↦ cmpt 4064 ⟶wf 5212 ‘cfv 5216 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 # cap 8537 ℤcz 9252 ℝ⁺crp 9652

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 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927

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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-inn 8919 df-z 9253 df-rp 9653

This theorem is referenced by: (None)

W3C validator