iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 34187

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 11177	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12540	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12584	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12586	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15495	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11255	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15534	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15363	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1577	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12339	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1577	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15334	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 705	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15373	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 705	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7435	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1577	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11232	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2789	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2817	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7439	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12585	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12368	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15470	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 3005	. . . . . . . 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 11980	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2817	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12538	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 34182	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 34182	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 34184	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11247	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12262	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12289	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 14	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11220	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11587	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15516	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12333	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15524	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12361	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11227	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 33127	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15519	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14251	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7441	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12403	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14253	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2792	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2817	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2801	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14314	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15533	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11660	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12386	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5158	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15500	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2817	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2811	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11574	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11587	. . . . . . . 8 ⊢ (⊤ → (-1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15516	. . . . . . 7 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15524	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(-1 − (i · (√‘3)))))
73	50	renegcld 11659	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 33127	. . . . . . . 8 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (-1↑2)))
75		neg1sqe1 14252	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (-1↑2) = 1)
77	52, 76	oveq12d 7441	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2817	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (2↑2))
79	74, 78	eqtrd 2801	. . . . . . 7 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14314	. . . . . 6 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15533	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(-1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2811	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 34179	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11667	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11884	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 34186	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2867	. 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 2146 ≠ wne 2961 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 ici 11120 + caddc 11121 · cmul 11123 ≤ cle 11262 − cmin 11459 -cneg 11460 / cdiv 11889 ℕcn 12251 2c2 12313 3c3 12314 4c4 12315 ℕ₀cn0 12522 ↑cexp 14117 √csqrt 15310 abscabs 15311 Constrcconstr 34150

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-constr 34151

This theorem is used by: constrremulcl 34188 constrmulcl 34192 constrreinvcl 34193 constrresqrtcl 34198 constrsqrtcl 34200

W3C validator