imsqrtvalex - Mathbox for Richard Penner

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Theorem imsqrtvalex 41143

Description: Example for imsqrtval 41141. (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 12179	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12183	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12381	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12175	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10861	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12000	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10913	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10912	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		imsqrtval 41141	. . 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 12015	. . . . 5 ⊢ 0 < 8
12		0re 10908	. . . . . . 7 ⊢ 0 ∈ ℝ
13		8re 11999	. . . . . . 7 ⊢ 8 ∈ ℝ
14	12, 13	ltnsymi 11024	. . . . . 6 ⊢ (0 < 8 → ¬ 8 < 0)
15	3	nn0rei 12174	. . . . . . . 8 ⊢ ;15 ∈ ℝ
16	15, 13	crimi 14832	. . . . . . 7 ⊢ (ℑ‘(;15 + (i · 8))) = 8
17	16	breq1i 5077	. . . . . 6 ⊢ ((ℑ‘(;15 + (i · 8))) < 0 ↔ 8 < 0)
18	14, 17	sylnibr 328	. . . . 5 ⊢ (0 < 8 → ¬ (ℑ‘(;15 + (i · 8))) < 0)
19	11, 18	ax-mp 5	. . . 4 ⊢ ¬ (ℑ‘(;15 + (i · 8))) < 0
20	19	iffalsei 4466	. . 3 ⊢ if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) = 1
21		absreim 14933	. . . . . . . . . . 11 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
22	15, 13, 21	mp2an 688	. . . . . . . . . 10 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
23	4	sqvali 13825	. . . . . . . . . . . . . 14 ⊢ (;15↑2) = (;15 · ;15)
24		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ;15 = ;15
25		7nn0 12185	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℕ₀
26	4	mulid2i 10911	. . . . . . . . . . . . . . . 16 ⊢ (1 · ;15) = ;15
27		1p1e2 12028	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
28		2nn0 12180	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
29	25	nn0cni 12175	. . . . . . . . . . . . . . . . 17 ⊢ 7 ∈ ℂ
30	2	nn0cni 12175	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
31		7p5e12 12443	. . . . . . . . . . . . . . . . 17 ⊢ (7 + 5) = ;12
32	29, 30, 31	addcomli 11097	. . . . . . . . . . . . . . . 16 ⊢ (5 + 7) = ;12
33	1, 2, 25, 26, 27, 28, 32	decaddci 12427	. . . . . . . . . . . . . . 15 ⊢ ((1 · ;15) + 7) = ;22
34	30	mulid1i 10910	. . . . . . . . . . . . . . . . . 18 ⊢ (5 · 1) = 5
35	34	oveq1i 7265	. . . . . . . . . . . . . . . . 17 ⊢ ((5 · 1) + 2) = (5 + 2)
36		5p2e7 12059	. . . . . . . . . . . . . . . . 17 ⊢ (5 + 2) = 7
37	35, 36	eqtri 2766	. . . . . . . . . . . . . . . 16 ⊢ ((5 · 1) + 2) = 7
38		5t5e25 12469	. . . . . . . . . . . . . . . 16 ⊢ (5 · 5) = ;25
39	2, 1, 2, 24, 2, 28, 37, 38	decmul2c 12432	. . . . . . . . . . . . . . 15 ⊢ (5 · ;15) = ;75
40	3, 1, 2, 24, 2, 25, 33, 39	decmul1c 12431	. . . . . . . . . . . . . 14 ⊢ (;15 · ;15) = ;;225
41	23, 40	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (;15↑2) = ;;225
42	6	sqvali 13825	. . . . . . . . . . . . . 14 ⊢ (8↑2) = (8 · 8)
43		8t8e64 12487	. . . . . . . . . . . . . 14 ⊢ (8 · 8) = ;64
44	42, 43	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (8↑2) = ;64
45	41, 44	oveq12i 7267	. . . . . . . . . . . 12 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
46	28, 28	deccl 12381	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ₀
47		6nn0 12184	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ₀
48		4nn0 12182	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
49		eqid 2738	. . . . . . . . . . . . 13 ⊢ ;;225 = ;;225
50		eqid 2738	. . . . . . . . . . . . 13 ⊢ ;64 = ;64
51		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
52	47	nn0cni 12175	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
53	28	nn0cni 12175	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
54		6p2e8 12062	. . . . . . . . . . . . . . 15 ⊢ (6 + 2) = 8
55	52, 53, 54	addcomli 11097	. . . . . . . . . . . . . 14 ⊢ (2 + 6) = 8
56	28, 28, 47, 51, 55	decaddi 12426	. . . . . . . . . . . . 13 ⊢ (;22 + 6) = ;28
57		5p4e9 12061	. . . . . . . . . . . . 13 ⊢ (5 + 4) = 9
58	46, 2, 47, 48, 49, 50, 56, 57	decadd 12420	. . . . . . . . . . . 12 ⊢ (;;225 + ;64) = ;;289
59	1, 25	deccl 12381	. . . . . . . . . . . . . . 15 ⊢ ;17 ∈ ℕ₀
60	59	nn0cni 12175	. . . . . . . . . . . . . 14 ⊢ ;17 ∈ ℂ
61	60	sqvali 13825	. . . . . . . . . . . . 13 ⊢ (;17↑2) = (;17 · ;17)
62		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ;17 = ;17
63		9nn0 12187	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
64	1, 1	deccl 12381	. . . . . . . . . . . . . 14 ⊢ ;11 ∈ ℕ₀
65	60	mulid2i 10911	. . . . . . . . . . . . . . 15 ⊢ (1 · ;17) = ;17
66		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ;11 = ;11
67		7p1e8 12052	. . . . . . . . . . . . . . 15 ⊢ (7 + 1) = 8
68	1, 25, 1, 1, 65, 66, 27, 67	decadd 12420	. . . . . . . . . . . . . 14 ⊢ ((1 · ;17) + ;11) = ;28
69	29	mulid1i 10910	. . . . . . . . . . . . . . . . 17 ⊢ (7 · 1) = 7
70	69	oveq1i 7265	. . . . . . . . . . . . . . . 16 ⊢ ((7 · 1) + 4) = (7 + 4)
71		7p4e11 12442	. . . . . . . . . . . . . . . 16 ⊢ (7 + 4) = ;11
72	70, 71	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ ((7 · 1) + 4) = ;11
73		7t7e49 12480	. . . . . . . . . . . . . . 15 ⊢ (7 · 7) = ;49
74	25, 1, 25, 62, 63, 48, 72, 73	decmul2c 12432	. . . . . . . . . . . . . 14 ⊢ (7 · ;17) = ;;119
75	59, 1, 25, 62, 63, 64, 68, 74	decmul1c 12431	. . . . . . . . . . . . 13 ⊢ (;17 · ;17) = ;;289
76	61, 75	eqtr2i 2767	. . . . . . . . . . . 12 ⊢ ;;289 = (;17↑2)
77	45, 58, 76	3eqtri 2770	. . . . . . . . . . 11 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
78	77	fveq2i 6759	. . . . . . . . . 10 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
79	59	nn0ge0i 12190	. . . . . . . . . . 11 ⊢ 0 ≤ ;17
80	59	nn0rei 12174	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℝ
81	80	sqrtsqi 15014	. . . . . . . . . . 11 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
82	79, 81	ax-mp 5	. . . . . . . . . 10 ⊢ (√‘(;17↑2)) = ;17
83	22, 78, 82	3eqtri 2770	. . . . . . . . 9 ⊢ (abs‘(;15 + (i · 8))) = ;17
84	15, 13	crrei 14831	. . . . . . . . 9 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
85	83, 84	oveq12i 7267	. . . . . . . 8 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = (;17 − ;15)
86	1, 2, 28, 24, 36	decaddi 12426	. . . . . . . . 9 ⊢ (;15 + 2) = ;17
87	60, 4, 53, 86	subaddrii 11240	. . . . . . . 8 ⊢ (;17 − ;15) = 2
88	85, 87	eqtri 2766	. . . . . . 7 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = 2
89	88	oveq1i 7265	. . . . . 6 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = (2 / 2)
90		2div2e1 12044	. . . . . 6 ⊢ (2 / 2) = 1
91	89, 90	eqtri 2766	. . . . 5 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = 1
92	91	fveq2i 6759	. . . 4 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘1)
93		sqrt1 14911	. . . 4 ⊢ (√‘1) = 1
94	92, 93	eqtri 2766	. . 3 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = 1
95	20, 94	oveq12i 7267	. 2 ⊢ (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))) = (1 · 1)
96		1t1e1 12065	. 2 ⊢ (1 · 1) = 1
97	10, 95, 96	3eqtri 2770	1 ⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2108 ifcif 4456 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 ici 10804 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 4c4 11960 5c5 11961 6c6 11962 7c7 11963 8c8 11964 9c9 11965 cdc 12366 ↑cexp 13710 ℜcre 14736 ℑcim 14737 √csqrt 14872 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: (None)

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