iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 34444

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 11262	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12627	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12671	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12673	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15590	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11340	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15629	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15458	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1576	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12426	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1576	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15429	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 705	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15468	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 705	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7432	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1576	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11317	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2783	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2811	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7436	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12672	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12455	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15565	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 2999	. . . . . . . 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 1576	. . . . 5 ⊢ (√‘3) ≠ 0
31	1, 19, 30	divcan4i 12067	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2811	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12625	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 34439	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 34439	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 34441	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11332	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12349	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12376	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 14	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11305	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11672	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15611	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12420	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15619	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12448	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11312	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 33383	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15614	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14341	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7438	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12490	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14343	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2786	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2811	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2795	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14405	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15628	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11747	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12473	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5146	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15595	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2811	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2805	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11659	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11672	. . . . . . . 8 ⊢ (⊤ → ( -1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15611	. . . . . . 7 ⊢ (⊤ → (abs‘( -1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15619	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘( -1 − (i · (√‘3)))))
73	50	renegcld 11746	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 33383	. . . . . . . 8 ⊢ (⊤ → ((abs‘( -1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + ( -1↑2)))
75		neg1sqe1 14342	. . . . . . . . . . 11 ⊢ ( -1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ( -1↑2) = 1)
77	52, 76	oveq12d 7438	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + ( -1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2811	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + ( -1↑2)) = (2↑2))
79	74, 78	eqtrd 2795	. . . . . . 7 ⊢ (⊤ → ((abs‘( -1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14405	. . . . . 6 ⊢ (⊤ → (abs‘( -1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15628	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘( -1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2805	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 34436	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11754	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11971	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 34443	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2861	. 2 ⊢ (⊤ → i ∈ Constr)
89	88	mptru 1576	1 ⊢ i ∈ Constr

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 ⊤wtru 1570 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 ici 11205 + caddc 11206 · cmul 11208 ≤ cle 11347 − cmin 11544 -cneg 11545 / cdiv 11976 ℕcn 12338 2c2 12400 3c3 12401 4c4 12402 ℕ₀cn0 12609 ↑cexp 14207 √csqrt 15405 abscabs 15406 Constrcconstr 34407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9437 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-n0 12610 df-z 12697 df-uz 12969 df-rp 13124 df-seq 14148 df-exp 14208 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-constr 34408

This theorem is used by: constrremulcl 34445 constrmulcl 34449 constrreinvcl 34450 constrresqrtcl 34455 constrsqrtcl 34457

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