iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 34284

Description: The imaginary unit i is constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)

**Assertion**
Ref	Expression
iconstr	⊢ i ∈ Constr

**Proof of Theorem iconstr**
Step	Hyp	Ref	Expression
1		ax-icn 11187	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12550	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12594	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12596	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15509	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11265	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15548	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15377	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1577	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12349	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1577	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15348	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 705	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15387	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 705	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7429	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1577	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11242	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2785	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2813	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7433	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12595	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12378	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15484	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 3001	. . . . . . . 8 ⊢ (3 ∈ ℂ → ((√‘3) ≠ 0 ↔ 3 ≠ 0))
28	27	biimpar 483	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (√‘3) ≠ 0)
29	24, 25, 28	sylancl 598	. . . . . 6 ⊢ (⊤ → (√‘3) ≠ 0)
30	29	mptru 1577	. . . . 5 ⊢ (√‘3) ≠ 0
31	1, 19, 30	divcan4i 11990	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2813	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12548	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 34279	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 34279	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 34281	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11257	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12272	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12299	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 14	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11230	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11597	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15530	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12343	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15538	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12371	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11237	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 33224	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15533	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14263	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7435	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12413	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14265	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2788	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2813	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2797	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14326	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15547	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11670	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12396	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5151	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15514	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2813	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2807	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11584	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11597	. . . . . . . 8 ⊢ (⊤ → (-1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15530	. . . . . . 7 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15538	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(-1 − (i · (√‘3)))))
73	50	renegcld 11669	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 33224	. . . . . . . 8 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (-1↑2)))
75		neg1sqe1 14264	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (-1↑2) = 1)
77	52, 76	oveq12d 7435	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2813	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (2↑2))
79	74, 78	eqtrd 2797	. . . . . . 7 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14326	. . . . . 6 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15547	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(-1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2807	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 34276	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11677	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11894	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 34283	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2863	. 2 ⊢ (⊤ → i ∈ Constr)
89	88	mptru 1577	1 ⊢ i ∈ Constr

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 ℝcr 11127 0cc0 11128 1c1 11129 ici 11130 + caddc 11131 · cmul 11133 ≤ cle 11272 − cmin 11469 -cneg 11470 / cdiv 11899 ℕcn 12261 2c2 12323 3c3 12324 4c4 12325 ℕ₀cn0 12532 ↑cexp 14129 √csqrt 15324 abscabs 15325 Constrcconstr 34247

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-constr 34248

This theorem is used by: constrremulcl 34285 constrmulcl 34289 constrreinvcl 34290 constrresqrtcl 34295 constrsqrtcl 34297

W3C validator