imsqrtvalex - Mathbox for Richard Penner

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Theorem imsqrtvalex 43608

Description: Example for imsqrtval 43606. (Contributed by RP, 21-May-2024.)

**Assertion**
Ref	Expression
imsqrtvalex	⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1

**Proof of Theorem imsqrtvalex**
Step	Hyp	Ref	Expression
1		1nn0 12569	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12573	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12773	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12565	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11243	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12390	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11297	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11296	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		imsqrtval 43606	. . 3 ⊢ ((;15 + (i · 8)) ∈ ℂ → (ℑ‘(√‘(;15 + (i · 8)))) = (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))))
10	8, 9	ax-mp 5	. 2 ⊢ (ℑ‘(√‘(;15 + (i · 8)))) = (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)))
11		8pos 12405	. . . . 5 ⊢ 0 < 8
12		0re 11292	. . . . . . 7 ⊢ 0 ∈ ℝ
13		8re 12389	. . . . . . 7 ⊢ 8 ∈ ℝ
14	12, 13	ltnsymi 11409	. . . . . 6 ⊢ (0 < 8 → ¬ 8 < 0)
15	3	nn0rei 12564	. . . . . . . 8 ⊢ ;15 ∈ ℝ
16	15, 13	crimi 15242	. . . . . . 7 ⊢ (ℑ‘(;15 + (i · 8))) = 8
17	16	breq1i 5173	. . . . . 6 ⊢ ((ℑ‘(;15 + (i · 8))) < 0 ↔ 8 < 0)
18	14, 17	sylnibr 329	. . . . 5 ⊢ (0 < 8 → ¬ (ℑ‘(;15 + (i · 8))) < 0)
19	11, 18	ax-mp 5	. . . 4 ⊢ ¬ (ℑ‘(;15 + (i · 8))) < 0
20	19	iffalsei 4558	. . 3 ⊢ if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) = 1
21		absreim 15342	. . . . . . . . . . 11 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
22	15, 13, 21	mp2an 691	. . . . . . . . . 10 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
23	4	sqvali 14229	. . . . . . . . . . . . . 14 ⊢ (;15↑2) = (;15 · ;15)
24		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ;15 = ;15
25		7nn0 12575	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℕ₀
26	4	mullidi 11295	. . . . . . . . . . . . . . . 16 ⊢ (1 · ;15) = ;15
27		1p1e2 12418	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
28		2nn0 12570	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
29	25	nn0cni 12565	. . . . . . . . . . . . . . . . 17 ⊢ 7 ∈ ℂ
30	2	nn0cni 12565	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
31		7p5e12 12835	. . . . . . . . . . . . . . . . 17 ⊢ (7 + 5) = ;12
32	29, 30, 31	addcomli 11482	. . . . . . . . . . . . . . . 16 ⊢ (5 + 7) = ;12
33	1, 2, 25, 26, 27, 28, 32	decaddci 12819	. . . . . . . . . . . . . . 15 ⊢ ((1 · ;15) + 7) = ;22
34	30	mulridi 11294	. . . . . . . . . . . . . . . . . 18 ⊢ (5 · 1) = 5
35	34	oveq1i 7458	. . . . . . . . . . . . . . . . 17 ⊢ ((5 · 1) + 2) = (5 + 2)
36		5p2e7 12449	. . . . . . . . . . . . . . . . 17 ⊢ (5 + 2) = 7
37	35, 36	eqtri 2768	. . . . . . . . . . . . . . . 16 ⊢ ((5 · 1) + 2) = 7
38		5t5e25 12861	. . . . . . . . . . . . . . . 16 ⊢ (5 · 5) = ;25
39	2, 1, 2, 24, 2, 28, 37, 38	decmul2c 12824	. . . . . . . . . . . . . . 15 ⊢ (5 · ;15) = ;75
40	3, 1, 2, 24, 2, 25, 33, 39	decmul1c 12823	. . . . . . . . . . . . . 14 ⊢ (;15 · ;15) = ;;225
41	23, 40	eqtri 2768	. . . . . . . . . . . . 13 ⊢ (;15↑2) = ;;225
42	6	sqvali 14229	. . . . . . . . . . . . . 14 ⊢ (8↑2) = (8 · 8)
43		8t8e64 12879	. . . . . . . . . . . . . 14 ⊢ (8 · 8) = ;64
44	42, 43	eqtri 2768	. . . . . . . . . . . . 13 ⊢ (8↑2) = ;64
45	41, 44	oveq12i 7460	. . . . . . . . . . . 12 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
46	28, 28	deccl 12773	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ₀
47		6nn0 12574	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ₀
48		4nn0 12572	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
49		eqid 2740	. . . . . . . . . . . . 13 ⊢ ;;225 = ;;225
50		eqid 2740	. . . . . . . . . . . . 13 ⊢ ;64 = ;64
51		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
52	47	nn0cni 12565	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
53	28	nn0cni 12565	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
54		6p2e8 12452	. . . . . . . . . . . . . . 15 ⊢ (6 + 2) = 8
55	52, 53, 54	addcomli 11482	. . . . . . . . . . . . . 14 ⊢ (2 + 6) = 8
56	28, 28, 47, 51, 55	decaddi 12818	. . . . . . . . . . . . 13 ⊢ (;22 + 6) = ;28
57		5p4e9 12451	. . . . . . . . . . . . 13 ⊢ (5 + 4) = 9
58	46, 2, 47, 48, 49, 50, 56, 57	decadd 12812	. . . . . . . . . . . 12 ⊢ (;;225 + ;64) = ;;289
59	1, 25	deccl 12773	. . . . . . . . . . . . . . 15 ⊢ ;17 ∈ ℕ₀
60	59	nn0cni 12565	. . . . . . . . . . . . . 14 ⊢ ;17 ∈ ℂ
61	60	sqvali 14229	. . . . . . . . . . . . 13 ⊢ (;17↑2) = (;17 · ;17)
62		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ;17 = ;17
63		9nn0 12577	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
64	1, 1	deccl 12773	. . . . . . . . . . . . . 14 ⊢ ;11 ∈ ℕ₀
65	60	mullidi 11295	. . . . . . . . . . . . . . 15 ⊢ (1 · ;17) = ;17
66		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ;11 = ;11
67		7p1e8 12442	. . . . . . . . . . . . . . 15 ⊢ (7 + 1) = 8
68	1, 25, 1, 1, 65, 66, 27, 67	decadd 12812	. . . . . . . . . . . . . 14 ⊢ ((1 · ;17) + ;11) = ;28
69	29	mulridi 11294	. . . . . . . . . . . . . . . . 17 ⊢ (7 · 1) = 7
70	69	oveq1i 7458	. . . . . . . . . . . . . . . 16 ⊢ ((7 · 1) + 4) = (7 + 4)
71		7p4e11 12834	. . . . . . . . . . . . . . . 16 ⊢ (7 + 4) = ;11
72	70, 71	eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ ((7 · 1) + 4) = ;11
73		7t7e49 12872	. . . . . . . . . . . . . . 15 ⊢ (7 · 7) = ;49
74	25, 1, 25, 62, 63, 48, 72, 73	decmul2c 12824	. . . . . . . . . . . . . 14 ⊢ (7 · ;17) = ;;119
75	59, 1, 25, 62, 63, 64, 68, 74	decmul1c 12823	. . . . . . . . . . . . 13 ⊢ (;17 · ;17) = ;;289
76	61, 75	eqtr2i 2769	. . . . . . . . . . . 12 ⊢ ;;289 = (;17↑2)
77	45, 58, 76	3eqtri 2772	. . . . . . . . . . 11 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
78	77	fveq2i 6923	. . . . . . . . . 10 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
79	59	nn0ge0i 12580	. . . . . . . . . . 11 ⊢ 0 ≤ ;17
80	59	nn0rei 12564	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℝ
81	80	sqrtsqi 15423	. . . . . . . . . . 11 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
82	79, 81	ax-mp 5	. . . . . . . . . 10 ⊢ (√‘(;17↑2)) = ;17
83	22, 78, 82	3eqtri 2772	. . . . . . . . 9 ⊢ (abs‘(;15 + (i · 8))) = ;17
84	15, 13	crrei 15241	. . . . . . . . 9 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
85	83, 84	oveq12i 7460	. . . . . . . 8 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = (;17 − ;15)
86	1, 2, 28, 24, 36	decaddi 12818	. . . . . . . . 9 ⊢ (;15 + 2) = ;17
87	60, 4, 53, 86	subaddrii 11625	. . . . . . . 8 ⊢ (;17 − ;15) = 2
88	85, 87	eqtri 2768	. . . . . . 7 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = 2
89	88	oveq1i 7458	. . . . . 6 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = (2 / 2)
90		2div2e1 12434	. . . . . 6 ⊢ (2 / 2) = 1
91	89, 90	eqtri 2768	. . . . 5 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = 1
92	91	fveq2i 6923	. . . 4 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘1)
93		sqrt1 15320	. . . 4 ⊢ (√‘1) = 1
94	92, 93	eqtri 2768	. . 3 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = 1
95	20, 94	oveq12i 7460	. 2 ⊢ (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))) = (1 · 1)
96		1t1e1 12455	. 2 ⊢ (1 · 1) = 1
97	10, 95, 96	3eqtri 2772	1 ⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2108 ifcif 4548 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 ici 11186 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 2c2 12348 4c4 12350 5c5 12351 6c6 12352 7c7 12353 8c8 12354 9c9 12355 cdc 12758 ↑cexp 14112 ℜcre 15146 ℑcim 15147 √csqrt 15282 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: (None)

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