constrrecl - Mathbox for Thierry Arnoux

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Theorem constrrecl 34104

Description: Constructible numbers are closed under taking the real part. (Contributed by Thierry Arnoux, 5-Nov-2025.)

**Hypothesis**
Ref	Expression
constrcjcl.1	⊢ (𝜑 → 𝑋 ∈ Constr)

**Assertion**
Ref	Expression
constrrecl	⊢ (𝜑 → (ℜ‘𝑋) ∈ Constr)

**Proof of Theorem constrrecl**
Step	Hyp	Ref	Expression
1		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → 𝑋 ∈ ℝ)
2	1	rered 15275	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = 𝑋)
3		constrcjcl.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Constr)
4	3	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → 𝑋 ∈ Constr)
5	2, 4	eqeltrd 2869	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ Constr)
6		0zd 12603	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
7	6	zconstr 34099	. . . 4 ⊢ (𝜑 → 0 ∈ Constr)
8	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 0 ∈ Constr)
9		1zzd 12625	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
10	9	zconstr 34099	. . . 4 ⊢ (𝜑 → 1 ∈ Constr)
11	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 1 ∈ Constr)
12	3	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 𝑋 ∈ Constr)
13	3	constrcjcl 34103	. . . 4 ⊢ (𝜑 → (∗‘𝑋) ∈ Constr)
14	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (∗‘𝑋) ∈ Constr)
15	3	constrcn 34095	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ)
16	15	recld 15245	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) ∈ ℝ)
17	16	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ ℝ)
18		halfre 12457	. . . 4 ⊢ (1 / 2) ∈ ℝ
19	18	a1i 11	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (1 / 2) ∈ ℝ)
20	16	recnd 11237	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) ∈ ℂ)
21	20	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ ℂ)
22		1cnd 11202	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
23	22	subid1d 11558	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
24	23, 22	eqeltrd 2869	. . . . . . 7 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
25	20, 24	mulcld 11229	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝑋) · (1 − 0)) ∈ ℂ)
26	25	addlidd 11411	. . . . 5 ⊢ (𝜑 → (0 + ((ℜ‘𝑋) · (1 − 0))) = ((ℜ‘𝑋) · (1 − 0)))
27	23	oveq2d 7427	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝑋) · (1 − 0)) = ((ℜ‘𝑋) · 1))
28	20	mulridd 11226	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝑋) · 1) = (ℜ‘𝑋))
29	26, 27, 28	3eqtrrd 2809	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) = (0 + ((ℜ‘𝑋) · (1 − 0))))
30	29	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = (0 + ((ℜ‘𝑋) · (1 − 0))))
31	15	cjcld 15247	. . . . . . 7 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
32	15, 31	addcld 11228	. . . . . 6 ⊢ (𝜑 → (𝑋 + (∗‘𝑋)) ∈ ℂ)
33		2cnd 12319	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
34		2ne0 12347	. . . . . . 7 ⊢ 2 ≠ 0
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
36	32, 33, 35	divrec2d 11995	. . . . 5 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) / 2) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
37		reval 15157	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2))
38	15, 37	syl 18	. . . . 5 ⊢ (𝜑 → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2))
39	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
40	39	recnd 11237	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
41	40, 31, 15	subdid 11670	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · ((∗‘𝑋) − 𝑋)) = (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋)))
42	41	oveq2d 7427	. . . . . 6 ⊢ (𝜑 → (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))) = (𝑋 + (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋))))
43	40, 15	mulcld 11229	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · 𝑋) ∈ ℂ)
44	40, 31	mulcld 11229	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · (∗‘𝑋)) ∈ ℂ)
45	15, 43, 44	subadd23d 11591	. . . . . 6 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = (𝑋 + (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋))))
46	22, 40, 15	subdird 11671	. . . . . . . . 9 ⊢ (𝜑 → ((1 − (1 / 2)) · 𝑋) = ((1 · 𝑋) − ((1 / 2) · 𝑋)))
47		1mhlfehlf 12463	. . . . . . . . . . 11 ⊢ (1 − (1 / 2)) = (1 / 2)
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (1 − (1 / 2)) = (1 / 2))
49	48	oveq1d 7426	. . . . . . . . 9 ⊢ (𝜑 → ((1 − (1 / 2)) · 𝑋) = ((1 / 2) · 𝑋))
50	15	mullidd 11227	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑋) = 𝑋)
51	50	oveq1d 7426	. . . . . . . . 9 ⊢ (𝜑 → ((1 · 𝑋) − ((1 / 2) · 𝑋)) = (𝑋 − ((1 / 2) · 𝑋)))
52	46, 49, 51	3eqtr3rd 2813	. . . . . . . 8 ⊢ (𝜑 → (𝑋 − ((1 / 2) · 𝑋)) = ((1 / 2) · 𝑋))
53	52	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = (((1 / 2) · 𝑋) + ((1 / 2) · (∗‘𝑋))))
54	40, 15, 31	adddid 11233	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · (𝑋 + (∗‘𝑋))) = (((1 / 2) · 𝑋) + ((1 / 2) · (∗‘𝑋))))
55	53, 54	eqtr4d 2807	. . . . . 6 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
56	42, 45, 55	3eqtr2d 2810	. . . . 5 ⊢ (𝜑 → (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
57	36, 38, 56	3eqtr4d 2814	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) = (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))))
58	57	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))))
59	23	fveq2d 6886	. . . . . . . . 9 ⊢ (𝜑 → (∗‘(1 − 0)) = (∗‘1))
60		1red 11209	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
61	60	cjred 15277	. . . . . . . . 9 ⊢ (𝜑 → (∗‘1) = 1)
62	59, 61	eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → (∗‘(1 − 0)) = 1)
63	62	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋)) = (1 · ((∗‘𝑋) − 𝑋)))
64	31, 15	subcld 11569	. . . . . . . 8 ⊢ (𝜑 → ((∗‘𝑋) − 𝑋) ∈ ℂ)
65	64	mullidd 11227	. . . . . . 7 ⊢ (𝜑 → (1 · ((∗‘𝑋) − 𝑋)) = ((∗‘𝑋) − 𝑋))
66	63, 65	eqtrd 2804	. . . . . 6 ⊢ (𝜑 → ((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋)) = ((∗‘𝑋) − 𝑋))
67	66	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) = (ℑ‘((∗‘𝑋) − 𝑋)))
68	67	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) = (ℑ‘((∗‘𝑋) − 𝑋)))
69	15	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → 𝑋 ∈ ℂ)
70		imval2 15202	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
71	15, 70	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
72	71	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
73	15, 31	subcld 11569	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 − (∗‘𝑋)) ∈ ℂ)
74	73	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) ∈ ℂ)
75	64	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → ((∗‘𝑋) − 𝑋) ∈ ℂ)
76	75	imnegd 15261	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘-((∗‘𝑋) − 𝑋)) = -(ℑ‘((∗‘𝑋) − 𝑋)))
77	31	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (∗‘𝑋) ∈ ℂ)
78	77, 69	negsubdi2d 11585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → -((∗‘𝑋) − 𝑋) = (𝑋 − (∗‘𝑋)))
79	78	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘-((∗‘𝑋) − 𝑋)) = (ℑ‘(𝑋 − (∗‘𝑋))))
80		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘((∗‘𝑋) − 𝑋)) = 0)
81	80	negeqd 11451	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → -(ℑ‘((∗‘𝑋) − 𝑋)) = -0)
82	76, 79, 81	3eqtr3d 2812	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘(𝑋 − (∗‘𝑋))) = -0)
83		neg0 11504	. . . . . . . . . . . . 13 ⊢ -0 = 0
84	82, 83	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘(𝑋 − (∗‘𝑋))) = 0)
85	74, 84	reim0bd 15251	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) ∈ ℝ)
86		cjth 15154	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → ((𝑋 + (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ))
87	15, 86	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ))
88	87	simprd 500	. . . . . . . . . . . 12 ⊢ (𝜑 → (i · (𝑋 − (∗‘𝑋))) ∈ ℝ)
89	88	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (i · (𝑋 − (∗‘𝑋))) ∈ ℝ)
90		rimul 12209	. . . . . . . . . . 11 ⊢ (((𝑋 − (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ) → (𝑋 − (∗‘𝑋)) = 0)
91	85, 89, 90	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) = 0)
92	91	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → ((𝑋 − (∗‘𝑋)) / (2 · i)) = (0 / (2 · i)))
93		ax-icn 11159	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
94	93	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → i ∈ ℂ)
95	33, 94	mulcld 11229	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · i) ∈ ℂ)
96	95	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (2 · i) ∈ ℂ)
97		ine0 11649	. . . . . . . . . . . . 13 ⊢ i ≠ 0
98	97	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → i ≠ 0)
99	33, 94, 35, 98	mulne0d 11866	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · i) ≠ 0)
100	99	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (2 · i) ≠ 0)
101	96, 100	div0d 11990	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (0 / (2 · i)) = 0)
102	72, 92, 101	3eqtrd 2808	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘𝑋) = 0)
103	69, 102	reim0bd 15251	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → 𝑋 ∈ ℝ)
104	103	ex 417	. . . . . 6 ⊢ (𝜑 → ((ℑ‘((∗‘𝑋) − 𝑋)) = 0 → 𝑋 ∈ ℝ))
105	104	necon3bd 2978	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 ∈ ℝ → (ℑ‘((∗‘𝑋) − 𝑋)) ≠ 0))
106	105	imp 411	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘𝑋) − 𝑋)) ≠ 0)
107	68, 106	eqnetrd 3031	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) ≠ 0)
108	8, 11, 12, 14, 17, 19, 21, 30, 58, 107	constrllcl 34091	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ Constr)
109	5, 108	pm2.61dan 824	1 ⊢ (𝜑 → (ℜ‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 ici 11102 + caddc 11103 · cmul 11105 − cmin 11441 -cneg 11442 / cdiv 11871 2c2 12295 ∗ccj 15147 ℜcre 15148 ℑcim 15149 Constrcconstr 34064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-n0 12505 df-z 12592 df-cj 15150 df-re 15151 df-im 15152 df-abs 15287 df-constr 34065

This theorem is referenced by: constrimcl 34105 constrmulcl 34106

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