constrrecl - Mathbox for Thierry Arnoux

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Theorem constrrecl 33754

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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → 𝑋 ∈ ℝ)
2	1	rered 15246	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = 𝑋)
3		constrcjcl.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Constr)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → 𝑋 ∈ Constr)
5	2, 4	eqeltrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ Constr)
6		0zd 12608	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
7	6	zconstr 33749	. . . 4 ⊢ (𝜑 → 0 ∈ Constr)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 0 ∈ Constr)
9		1zzd 12631	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
10	9	zconstr 33749	. . . 4 ⊢ (𝜑 → 1 ∈ Constr)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 1 ∈ Constr)
12	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → 𝑋 ∈ Constr)
13	3	constrcjcl 33753	. . . 4 ⊢ (𝜑 → (∗‘𝑋) ∈ Constr)
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (∗‘𝑋) ∈ Constr)
15	3	constrcn 33745	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ)
16	15	recld 15216	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) ∈ ℝ)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ ℝ)
18		halfre 12462	. . . 4 ⊢ (1 / 2) ∈ ℝ
19	18	a1i 11	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (1 / 2) ∈ ℝ)
20	16	recnd 11271	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) ∈ ℂ)
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ ℂ)
22		1cnd 11238	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
23	22	subid1d 11591	. . . . . . . 8 ⊢ (𝜑 → (1 − 0) = 1)
24	23, 22	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → (1 − 0) ∈ ℂ)
25	20, 24	mulcld 11263	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝑋) · (1 − 0)) ∈ ℂ)
26	25	addlidd 11444	. . . . 5 ⊢ (𝜑 → (0 + ((ℜ‘𝑋) · (1 − 0))) = ((ℜ‘𝑋) · (1 − 0)))
27	23	oveq2d 7429	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝑋) · (1 − 0)) = ((ℜ‘𝑋) · 1))
28	20	mulridd 11260	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝑋) · 1) = (ℜ‘𝑋))
29	26, 27, 28	3eqtrrd 2774	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) = (0 + ((ℜ‘𝑋) · (1 − 0))))
30	29	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = (0 + ((ℜ‘𝑋) · (1 − 0))))
31	15	cjcld 15218	. . . . . . 7 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ)
32	15, 31	addcld 11262	. . . . . 6 ⊢ (𝜑 → (𝑋 + (∗‘𝑋)) ∈ ℂ)
33		2cnd 12326	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
34		2ne0 12352	. . . . . . 7 ⊢ 2 ≠ 0
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
36	32, 33, 35	divrec2d 12029	. . . . 5 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) / 2) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
37		reval 15128	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2))
38	15, 37	syl 17	. . . . 5 ⊢ (𝜑 → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2))
39	18	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
40	39	recnd 11271	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
41	40, 31, 15	subdid 11701	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · ((∗‘𝑋) − 𝑋)) = (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋)))
42	41	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))) = (𝑋 + (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋))))
43	40, 15	mulcld 11263	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · 𝑋) ∈ ℂ)
44	40, 31	mulcld 11263	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · (∗‘𝑋)) ∈ ℂ)
45	15, 43, 44	subadd23d 11624	. . . . . 6 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = (𝑋 + (((1 / 2) · (∗‘𝑋)) − ((1 / 2) · 𝑋))))
46	22, 40, 15	subdird 11702	. . . . . . . . 9 ⊢ (𝜑 → ((1 − (1 / 2)) · 𝑋) = ((1 · 𝑋) − ((1 / 2) · 𝑋)))
47		1mhlfehlf 12468	. . . . . . . . . . 11 ⊢ (1 − (1 / 2)) = (1 / 2)
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (1 − (1 / 2)) = (1 / 2))
49	48	oveq1d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((1 − (1 / 2)) · 𝑋) = ((1 / 2) · 𝑋))
50	15	mullidd 11261	. . . . . . . . . 10 ⊢ (𝜑 → (1 · 𝑋) = 𝑋)
51	50	oveq1d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((1 · 𝑋) − ((1 / 2) · 𝑋)) = (𝑋 − ((1 / 2) · 𝑋)))
52	46, 49, 51	3eqtr3rd 2778	. . . . . . . 8 ⊢ (𝜑 → (𝑋 − ((1 / 2) · 𝑋)) = ((1 / 2) · 𝑋))
53	52	oveq1d 7428	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = (((1 / 2) · 𝑋) + ((1 / 2) · (∗‘𝑋))))
54	40, 15, 31	adddid 11267	. . . . . . 7 ⊢ (𝜑 → ((1 / 2) · (𝑋 + (∗‘𝑋))) = (((1 / 2) · 𝑋) + ((1 / 2) · (∗‘𝑋))))
55	53, 54	eqtr4d 2772	. . . . . 6 ⊢ (𝜑 → ((𝑋 − ((1 / 2) · 𝑋)) + ((1 / 2) · (∗‘𝑋))) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
56	42, 45, 55	3eqtr2d 2775	. . . . 5 ⊢ (𝜑 → (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))) = ((1 / 2) · (𝑋 + (∗‘𝑋))))
57	36, 38, 56	3eqtr4d 2779	. . . 4 ⊢ (𝜑 → (ℜ‘𝑋) = (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))))
58	57	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) = (𝑋 + ((1 / 2) · ((∗‘𝑋) − 𝑋))))
59	23	fveq2d 6890	. . . . . . . . 9 ⊢ (𝜑 → (∗‘(1 − 0)) = (∗‘1))
60		1red 11244	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
61	60	cjred 15248	. . . . . . . . 9 ⊢ (𝜑 → (∗‘1) = 1)
62	59, 61	eqtrd 2769	. . . . . . . 8 ⊢ (𝜑 → (∗‘(1 − 0)) = 1)
63	62	oveq1d 7428	. . . . . . 7 ⊢ (𝜑 → ((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋)) = (1 · ((∗‘𝑋) − 𝑋)))
64	31, 15	subcld 11602	. . . . . . . 8 ⊢ (𝜑 → ((∗‘𝑋) − 𝑋) ∈ ℂ)
65	64	mullidd 11261	. . . . . . 7 ⊢ (𝜑 → (1 · ((∗‘𝑋) − 𝑋)) = ((∗‘𝑋) − 𝑋))
66	63, 65	eqtrd 2769	. . . . . 6 ⊢ (𝜑 → ((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋)) = ((∗‘𝑋) − 𝑋))
67	66	fveq2d 6890	. . . . 5 ⊢ (𝜑 → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) = (ℑ‘((∗‘𝑋) − 𝑋)))
68	67	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) = (ℑ‘((∗‘𝑋) − 𝑋)))
69	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → 𝑋 ∈ ℂ)
70		imval2 15173	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
71	15, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
72	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘𝑋) = ((𝑋 − (∗‘𝑋)) / (2 · i)))
73	15, 31	subcld 11602	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 − (∗‘𝑋)) ∈ ℂ)
74	73	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) ∈ ℂ)
75	64	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → ((∗‘𝑋) − 𝑋) ∈ ℂ)
76	75	imnegd 15232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘-((∗‘𝑋) − 𝑋)) = -(ℑ‘((∗‘𝑋) − 𝑋)))
77	31	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (∗‘𝑋) ∈ ℂ)
78	77, 69	negsubdi2d 11618	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → -((∗‘𝑋) − 𝑋) = (𝑋 − (∗‘𝑋)))
79	78	fveq2d 6890	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘-((∗‘𝑋) − 𝑋)) = (ℑ‘(𝑋 − (∗‘𝑋))))
80		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘((∗‘𝑋) − 𝑋)) = 0)
81	80	negeqd 11484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → -(ℑ‘((∗‘𝑋) − 𝑋)) = -0)
82	76, 79, 81	3eqtr3d 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘(𝑋 − (∗‘𝑋))) = -0)
83		neg0 11537	. . . . . . . . . . . . 13 ⊢ -0 = 0
84	82, 83	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘(𝑋 − (∗‘𝑋))) = 0)
85	74, 84	reim0bd 15222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) ∈ ℝ)
86		cjth 15125	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → ((𝑋 + (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ))
87	15, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ))
88	87	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → (i · (𝑋 − (∗‘𝑋))) ∈ ℝ)
89	88	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (i · (𝑋 − (∗‘𝑋))) ∈ ℝ)
90		rimul 12239	. . . . . . . . . . 11 ⊢ (((𝑋 − (∗‘𝑋)) ∈ ℝ ∧ (i · (𝑋 − (∗‘𝑋))) ∈ ℝ) → (𝑋 − (∗‘𝑋)) = 0)
91	85, 89, 90	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (𝑋 − (∗‘𝑋)) = 0)
92	91	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → ((𝑋 − (∗‘𝑋)) / (2 · i)) = (0 / (2 · i)))
93		ax-icn 11196	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
94	93	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → i ∈ ℂ)
95	33, 94	mulcld 11263	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · i) ∈ ℂ)
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (2 · i) ∈ ℂ)
97		ine0 11680	. . . . . . . . . . . . 13 ⊢ i ≠ 0
98	97	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → i ≠ 0)
99	33, 94, 35, 98	mulne0d 11897	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · i) ≠ 0)
100	99	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (2 · i) ≠ 0)
101	96, 100	div0d 12024	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (0 / (2 · i)) = 0)
102	72, 92, 101	3eqtrd 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → (ℑ‘𝑋) = 0)
103	69, 102	reim0bd 15222	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘((∗‘𝑋) − 𝑋)) = 0) → 𝑋 ∈ ℝ)
104	103	ex 412	. . . . . 6 ⊢ (𝜑 → ((ℑ‘((∗‘𝑋) − 𝑋)) = 0 → 𝑋 ∈ ℝ))
105	104	necon3bd 2945	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 ∈ ℝ → (ℑ‘((∗‘𝑋) − 𝑋)) ≠ 0))
106	105	imp 406	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘𝑋) − 𝑋)) ≠ 0)
107	68, 106	eqnetrd 2998	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℑ‘((∗‘(1 − 0)) · ((∗‘𝑋) − 𝑋))) ≠ 0)
108	8, 11, 12, 14, 17, 19, 21, 30, 58, 107	constrllcl 33741	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ℝ) → (ℜ‘𝑋) ∈ Constr)
109	5, 108	pm2.61dan 812	1 ⊢ (𝜑 → (ℜ‘𝑋) ∈ Constr)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ‘cfv 6541 (class class class)co 7413 ℂcc 11135 ℝcr 11136 0cc0 11137 1c1 11138 ici 11139 + caddc 11140 · cmul 11142 − cmin 11474 -cneg 11475 / cdiv 11902 2c2 12303 ∗ccj 15118 ℜcre 15119 ℑcim 15120 Constrcconstr 33714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-n0 12510 df-z 12597 df-cj 15121 df-re 15122 df-im 15123 df-abs 15258 df-constr 33715

This theorem is referenced by: constrimcl 33755 constrmulcl 33756

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