iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 33800

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 11072	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12406	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12450	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12452	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15327	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11147	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15366	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15195	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1548	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12212	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1548	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15166	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 692	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15205	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 692	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7364	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1548	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11124	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2756	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2784	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7368	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12451	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12238	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15302	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 2973	. . . . . . . 8 ⊢ (3 ∈ ℂ → ((√‘3) ≠ 0 ↔ 3 ≠ 0))
28	27	biimpar 477	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (√‘3) ≠ 0)
29	24, 25, 28	sylancl 586	. . . . . 6 ⊢ (⊤ → (√‘3) ≠ 0)
30	29	mptru 1548	. . . . 5 ⊢ (√‘3) ≠ 0
31	1, 19, 30	divcan4i 11875	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2784	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12404	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 33795	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 33795	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 33797	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11139	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12143	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12167	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 13	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11114	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11479	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15348	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12206	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15356	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12234	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11120	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 32733	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15351	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14104	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7370	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12272	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14106	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2759	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2784	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2768	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14167	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15365	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11552	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12255	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5135	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15332	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2784	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2778	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11466	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11479	. . . . . . . 8 ⊢ (⊤ → (-1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15348	. . . . . . 7 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15356	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(-1 − (i · (√‘3)))))
73	50	renegcld 11551	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 32733	. . . . . . . 8 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (-1↑2)))
75		neg1sqe1 14105	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (-1↑2) = 1)
77	52, 76	oveq12d 7370	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2784	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (2↑2))
79	74, 78	eqtrd 2768	. . . . . . 7 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14167	. . . . . 6 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15365	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(-1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2778	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 33792	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11559	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11776	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 33799	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2834	. 2 ⊢ (⊤ → i ∈ Constr)
89	88	mptru 1548	1 ⊢ i ∈ Constr

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2929 class class class wbr 5093 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 ici 11015 + caddc 11016 · cmul 11018 ≤ cle 11154 − cmin 11351 -cneg 11352 / cdiv 11781 ℕcn 12132 2c2 12187 3c3 12188 4c4 12189 ℕ₀cn0 12388 ↑cexp 13970 √csqrt 15142 abscabs 15143 Constrcconstr 33763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-constr 33764

This theorem is referenced by: constrremulcl 33801 constrmulcl 33805 constrreinvcl 33806 constrresqrtcl 33811 constrsqrtcl 33813

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