iconstr - Mathbox for Thierry Arnoux

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Theorem iconstr 34380

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 11240	. . . . . . . 8 ⊢ i ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
3		3nn0 12605	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℕ₀)
5	4	nn0red 12649	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℝ)
6	4	nn0ge0d 12651	. . . . . . . . 9 ⊢ (⊤ → 0 ≤ 3)
7	5, 6	resqrtcld 15565	. . . . . . . 8 ⊢ (⊤ → (√‘3) ∈ ℝ)
8	7	recnd 11318	. . . . . . 7 ⊢ (⊤ → (√‘3) ∈ ℂ)
9	2, 8	absmuld 15604	. . . . . 6 ⊢ (⊤ → (abs‘(i · (√‘3))) = ((abs‘i) · (abs‘(√‘3))))
10		absi 15433	. . . . . . . 8 ⊢ (abs‘i) = 1
11	7	mptru 1577	. . . . . . . . 9 ⊢ (√‘3) ∈ ℝ
12		3re 12404	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
13	6	mptru 1577	. . . . . . . . . 10 ⊢ 0 ≤ 3
14		sqrtge0 15404	. . . . . . . . . 10 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → 0 ≤ (√‘3))
15	12, 13, 14	mp2an 705	. . . . . . . . 9 ⊢ 0 ≤ (√‘3)
16		absid 15443	. . . . . . . . 9 ⊢ (((√‘3) ∈ ℝ ∧ 0 ≤ (√‘3)) → (abs‘(√‘3)) = (√‘3))
17	11, 15, 16	mp2an 705	. . . . . . . 8 ⊢ (abs‘(√‘3)) = (√‘3)
18	10, 17	oveq12i 7424	. . . . . . 7 ⊢ ((abs‘i) · (abs‘(√‘3))) = (1 · (√‘3))
19	8	mptru 1577	. . . . . . . 8 ⊢ (√‘3) ∈ ℂ
20	19	mullidi 11295	. . . . . . 7 ⊢ (1 · (√‘3)) = (√‘3)
21	18, 20	eqtri 2784	. . . . . 6 ⊢ ((abs‘i) · (abs‘(√‘3))) = (√‘3)
22	9, 21	eqtrdi 2812	. . . . 5 ⊢ (⊤ → (abs‘(i · (√‘3))) = (√‘3))
23	22	oveq2d 7428	. . . 4 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = ((i · (√‘3)) / (√‘3)))
24	4	nn0cnd 12650	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℂ)
25		3ne0 12433	. . . . . . 7 ⊢ 3 ≠ 0
26		cnsqrt00 15540	. . . . . . . . 9 ⊢ (3 ∈ ℂ → ((√‘3) = 0 ↔ 3 = 0))
27	26	necon3bid 3000	. . . . . . . 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 12045	. . . 4 ⊢ ((i · (√‘3)) / (√‘3)) = i
32	23, 31	eqtrdi 2812	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) = i)
33		1nn0 12603	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34	33	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ ℕ₀)
35	34	nn0constr 34375	. . . . 5 ⊢ (⊤ → 1 ∈ Constr)
36	4	nn0constr 34375	. . . . 5 ⊢ (⊤ → 3 ∈ Constr)
37	35	constrnegcl 34377	. . . . 5 ⊢ (⊤ → -1 ∈ Constr)
38	2, 8	mulcld 11310	. . . . 5 ⊢ (⊤ → (i · (√‘3)) ∈ ℂ)
39		1nn 12327	. . . . . 6 ⊢ 1 ∈ ℕ
40		nnneneg 12354	. . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1)
41	39, 40	mp1i 14	. . . . 5 ⊢ (⊤ → 1 ≠ -1)
42		1cnd 11283	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
43	42, 38	subcld 11650	. . . . . . . 8 ⊢ (⊤ → (1 − (i · (√‘3))) ∈ ℂ)
44	43	abscld 15586	. . . . . . 7 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) ∈ ℝ)
45		2re 12398	. . . . . . . 8 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℝ)
47	43	absge0d 15594	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(1 − (i · (√‘3)))))
48		0le2 12426	. . . . . . . 8 ⊢ 0 ≤ 2
49	48	a1i 11	. . . . . . 7 ⊢ (⊤ → 0 ≤ 2)
50		1red 11290	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℝ)
51	7, 50	pythagreim 33319	. . . . . . . 8 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (1↑2)))
52	24	sqsqrtd 15589	. . . . . . . . . 10 ⊢ (⊤ → ((√‘3)↑2) = 3)
53		sq1 14318	. . . . . . . . . . 11 ⊢ (1↑2) = 1
54	53	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (1↑2) = 1)
55	52, 54	oveq12d 7430	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (3 + 1))
56		3p1e4 12468	. . . . . . . . . 10 ⊢ (3 + 1) = 4
57		sq2 14320	. . . . . . . . . 10 ⊢ (2↑2) = 4
58	56, 57	eqtr4i 2787	. . . . . . . . 9 ⊢ (3 + 1) = (2↑2)
59	55, 58	eqtrdi 2812	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (1↑2)) = (2↑2))
60	51, 59	eqtrd 2796	. . . . . . 7 ⊢ (⊤ → ((abs‘(1 − (i · (√‘3))))↑2) = (2↑2))
61	44, 46, 47, 49, 60	sq11d 14382	. . . . . 6 ⊢ (⊤ → (abs‘(1 − (i · (√‘3)))) = 2)
62	38, 42	abssubd 15603	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(1 − (i · (√‘3)))))
63	5, 50	resubcld 11725	. . . . . . . 8 ⊢ (⊤ → (3 − 1) ∈ ℝ)
64		3m1e2 12451	. . . . . . . . 9 ⊢ (3 − 1) = 2
65	49, 64	breqtrrdi 5147	. . . . . . . 8 ⊢ (⊤ → 0 ≤ (3 − 1))
66	63, 65	absidd 15570	. . . . . . 7 ⊢ (⊤ → (abs‘(3 − 1)) = (3 − 1))
67	66, 64	eqtrdi 2812	. . . . . 6 ⊢ (⊤ → (abs‘(3 − 1)) = 2)
68	61, 62, 67	3eqtr4d 2806	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − 1)) = (abs‘(3 − 1)))
69	42	negcld 11637	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℂ)
70	69, 38	subcld 11650	. . . . . . . 8 ⊢ (⊤ → (-1 − (i · (√‘3))) ∈ ℂ)
71	70	abscld 15586	. . . . . . 7 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) ∈ ℝ)
72	70	absge0d 15594	. . . . . . 7 ⊢ (⊤ → 0 ≤ (abs‘(-1 − (i · (√‘3)))))
73	50	renegcld 11724	. . . . . . . . 9 ⊢ (⊤ → -1 ∈ ℝ)
74	7, 73	pythagreim 33319	. . . . . . . 8 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (((√‘3)↑2) + (-1↑2)))
75		neg1sqe1 14319	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
76	75	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (-1↑2) = 1)
77	52, 76	oveq12d 7430	. . . . . . . . 9 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (3 + 1))
78	77, 58	eqtrdi 2812	. . . . . . . 8 ⊢ (⊤ → (((√‘3)↑2) + (-1↑2)) = (2↑2))
79	74, 78	eqtrd 2796	. . . . . . 7 ⊢ (⊤ → ((abs‘(-1 − (i · (√‘3))))↑2) = (2↑2))
80	71, 46, 72, 49, 79	sq11d 14382	. . . . . 6 ⊢ (⊤ → (abs‘(-1 − (i · (√‘3)))) = 2)
81	38, 69	abssubd 15603	. . . . . 6 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(-1 − (i · (√‘3)))))
82	80, 81, 67	3eqtr4d 2806	. . . . 5 ⊢ (⊤ → (abs‘((i · (√‘3)) − -1)) = (abs‘(3 − 1)))
83	35, 36, 35, 37, 36, 35, 38, 41, 68, 82	constrcccl 34372	. . . 4 ⊢ (⊤ → (i · (√‘3)) ∈ Constr)
84		ine0 11732	. . . . . 6 ⊢ i ≠ 0
85	84	a1i 11	. . . . 5 ⊢ (⊤ → i ≠ 0)
86	2, 8, 85, 29	mulne0d 11949	. . . 4 ⊢ (⊤ → (i · (√‘3)) ≠ 0)
87	83, 86	constrdircl 34379	. . 3 ⊢ (⊤ → ((i · (√‘3)) / (abs‘(i · (√‘3)))) ∈ Constr)
88	32, 87	eqeltrrd 2862	. 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 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 ici 11183 + caddc 11184 · cmul 11186 ≤ cle 11325 − cmin 11522 -cneg 11523 / cdiv 11954 ℕcn 12316 2c2 12378 3c3 12379 4c4 12380 ℕ₀cn0 12587 ↑cexp 14184 √csqrt 15380 abscabs 15381 Constrcconstr 34343

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9418 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-n0 12588 df-z 12675 df-uz 12947 df-rp 13102 df-seq 14125 df-exp 14185 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-constr 34344

This theorem is used by: constrremulcl 34381 constrmulcl 34385 constrreinvcl 34386 constrresqrtcl 34391 constrsqrtcl 34393

W3C validator