iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 33926

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 11088	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12446	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12490	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12492	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15371	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11164	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15410	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15239	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1549	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12252	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1549	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15210	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 693	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15249	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 693	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7372	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1549	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11141	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2760	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2788	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7376	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12491	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12278	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15346	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 2977	. . . . . . . 8 ⊢ (3 ∈ ℂ → ((√‘3) ≠ 0 ↔ 3 ≠ 0))
28	27	biimpar 477	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (√‘3) ≠ 0)
29	24, 25, 28	sylancl 587	. . . . . 6 ⊢ (⊤ → (√‘3) ≠ 0)
30	29	mptru 1549	. . . . 5 ⊢ (√‘3) ≠ 0
31	1, 19, 30	divcan4i 11893	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2788	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12444	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 33921	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 33921	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 33923	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11156	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12176	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12203	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 13	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11130	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11496	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15392	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12246	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15400	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12274	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11136	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 32833	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15395	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14148	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7378	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12312	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14150	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2763	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2788	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2772	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14211	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15409	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11569	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12295	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5128	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15376	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2788	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2782	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11483	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11496	. . . . . . . 8 ⊢ (⊤ → (-1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15392	. . . . . . 7 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15400	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(-1 − (i · (√‘3)))))
73	50	renegcld 11568	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 32833	. . . . . . . 8 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (-1↑2)))
75		neg1sqe1 14149	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (-1↑2) = 1)
77	52, 76	oveq12d 7378	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2788	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (2↑2))
79	74, 78	eqtrd 2772	. . . . . . 7 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14211	. . . . . 6 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15409	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(-1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2782	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 33918	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11576	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11793	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 33925	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2838	. 2 ⊢ (⊤ → i ∈ Constr)
89	88	mptru 1549	1 ⊢ i ∈ Constr

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 ici 11031 + caddc 11032 · cmul 11034 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 ℕcn 12165 2c2 12227 3c3 12228 4c4 12229 ℕ₀cn0 12428 ↑cexp 14014 √csqrt 15186 abscabs 15187 Constrcconstr 33889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-constr 33890

This theorem is referenced by: constrremulcl 33927 constrmulcl 33931 constrreinvcl 33932 constrresqrtcl 33937 constrsqrtcl 33939

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