imsqrtvalex - Mathbox for Richard Penner

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Theorem imsqrtvalex 44320

Description: Example for imsqrtval 44318. (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 12519	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12523	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12725	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12515	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11158	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12337	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11215	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11214	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		imsqrtval 44318	. . 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 12355	. . . . 5 ⊢ 0 < 8
12		0re 11209	. . . . . . 7 ⊢ 0 ∈ ℝ
13		8re 12336	. . . . . . 7 ⊢ 8 ∈ ℝ
14	12, 13	ltnsymi 11328	. . . . . 6 ⊢ (0 < 8 → ¬ 8 < 0)
15	3	nn0rei 12514	. . . . . . . 8 ⊢ ;15 ∈ ℝ
16	15, 13	crimi 15243	. . . . . . 7 ⊢ (ℑ‘(;15 + (i · 8))) = 8
17	16	breq1i 5115	. . . . . 6 ⊢ ((ℑ‘(;15 + (i · 8))) < 0 ↔ 8 < 0)
18	14, 17	sylnibr 332	. . . . 5 ⊢ (0 < 8 → ¬ (ℑ‘(;15 + (i · 8))) < 0)
19	11, 18	ax-mp 5	. . . 4 ⊢ ¬ (ℑ‘(;15 + (i · 8))) < 0
20	19	iffalsei 4496	. . 3 ⊢ if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) = 1
21		absreim 15343	. . . . . . . . . . 11 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
22	15, 13, 21	mp2an 704	. . . . . . . . . 10 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
23	4	sqvali 14215	. . . . . . . . . . . . . 14 ⊢ (;15↑2) = (;15 · ;15)
24		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ ;15 = ;15
25		7nn0 12525	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℕ₀
26	4	mullidi 11213	. . . . . . . . . . . . . . . 16 ⊢ (1 · ;15) = ;15
27		1p1e2 12363	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
28		2nn0 12520	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
29	25	nn0cni 12515	. . . . . . . . . . . . . . . . 17 ⊢ 7 ∈ ℂ
30	2	nn0cni 12515	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
31		7p5e12 12792	. . . . . . . . . . . . . . . . 17 ⊢ (7 + 5) = ;12
32	29, 30, 31	addcomli 11401	. . . . . . . . . . . . . . . 16 ⊢ (5 + 7) = ;12
33	1, 2, 25, 26, 27, 28, 32	decaddci 12776	. . . . . . . . . . . . . . 15 ⊢ ((1 · ;15) + 7) = ;22
34	30	mulridi 11212	. . . . . . . . . . . . . . . . . 18 ⊢ (5 · 1) = 5
35	34	oveq1i 7420	. . . . . . . . . . . . . . . . 17 ⊢ ((5 · 1) + 2) = (5 + 2)
36		5p2e7 12395	. . . . . . . . . . . . . . . . 17 ⊢ (5 + 2) = 7
37	35, 36	eqtri 2784	. . . . . . . . . . . . . . . 16 ⊢ ((5 · 1) + 2) = 7
38		5t5e25 12818	. . . . . . . . . . . . . . . 16 ⊢ (5 · 5) = ;25
39	2, 1, 2, 24, 2, 28, 37, 38	decmul2c 12781	. . . . . . . . . . . . . . 15 ⊢ (5 · ;15) = ;75
40	3, 1, 2, 24, 2, 25, 33, 39	decmul1c 12780	. . . . . . . . . . . . . 14 ⊢ (;15 · ;15) = ;;225
41	23, 40	eqtri 2784	. . . . . . . . . . . . 13 ⊢ (;15↑2) = ;;225
42	6	sqvali 14215	. . . . . . . . . . . . . 14 ⊢ (8↑2) = (8 · 8)
43		8t8e64 12836	. . . . . . . . . . . . . 14 ⊢ (8 · 8) = ;64
44	42, 43	eqtri 2784	. . . . . . . . . . . . 13 ⊢ (8↑2) = ;64
45	41, 44	oveq12i 7422	. . . . . . . . . . . 12 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
46	28, 28	deccl 12725	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ₀
47		6nn0 12524	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ₀
48		4nn0 12522	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
49		eqid 2761	. . . . . . . . . . . . 13 ⊢ ;;225 = ;;225
50		eqid 2761	. . . . . . . . . . . . 13 ⊢ ;64 = ;64
51		eqid 2761	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
52	47	nn0cni 12515	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
53	28	nn0cni 12515	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
54		6p2e8 12398	. . . . . . . . . . . . . . 15 ⊢ (6 + 2) = 8
55	52, 53, 54	addcomli 11401	. . . . . . . . . . . . . 14 ⊢ (2 + 6) = 8
56	28, 28, 47, 51, 55	decaddi 12775	. . . . . . . . . . . . 13 ⊢ (;22 + 6) = ;28
57		5p4e9 12397	. . . . . . . . . . . . 13 ⊢ (5 + 4) = 9
58	46, 2, 47, 48, 49, 50, 56, 57	decadd 12769	. . . . . . . . . . . 12 ⊢ (;;225 + ;64) = ;;289
59	1, 25	deccl 12725	. . . . . . . . . . . . . . 15 ⊢ ;17 ∈ ℕ₀
60	59	nn0cni 12515	. . . . . . . . . . . . . 14 ⊢ ;17 ∈ ℂ
61	60	sqvali 14215	. . . . . . . . . . . . 13 ⊢ (;17↑2) = (;17 · ;17)
62		eqid 2761	. . . . . . . . . . . . . 14 ⊢ ;17 = ;17
63		9nn0 12527	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
64	1, 1	deccl 12725	. . . . . . . . . . . . . 14 ⊢ ;11 ∈ ℕ₀
65	60	mullidi 11213	. . . . . . . . . . . . . . 15 ⊢ (1 · ;17) = ;17
66		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ ;11 = ;11
67		7p1e8 12388	. . . . . . . . . . . . . . 15 ⊢ (7 + 1) = 8
68	1, 25, 1, 1, 65, 66, 27, 67	decadd 12769	. . . . . . . . . . . . . 14 ⊢ ((1 · ;17) + ;11) = ;28
69	29	mulridi 11212	. . . . . . . . . . . . . . . . 17 ⊢ (7 · 1) = 7
70	69	oveq1i 7420	. . . . . . . . . . . . . . . 16 ⊢ ((7 · 1) + 4) = (7 + 4)
71		7p4e11 12791	. . . . . . . . . . . . . . . 16 ⊢ (7 + 4) = ;11
72	70, 71	eqtri 2784	. . . . . . . . . . . . . . 15 ⊢ ((7 · 1) + 4) = ;11
73		7t7e49 12829	. . . . . . . . . . . . . . 15 ⊢ (7 · 7) = ;49
74	25, 1, 25, 62, 63, 48, 72, 73	decmul2c 12781	. . . . . . . . . . . . . 14 ⊢ (7 · ;17) = ;;119
75	59, 1, 25, 62, 63, 64, 68, 74	decmul1c 12780	. . . . . . . . . . . . 13 ⊢ (;17 · ;17) = ;;289
76	61, 75	eqtr2i 2785	. . . . . . . . . . . 12 ⊢ ;;289 = (;17↑2)
77	45, 58, 76	3eqtri 2788	. . . . . . . . . . 11 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
78	77	fveq2i 6884	. . . . . . . . . 10 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
79	59	nn0ge0i 12530	. . . . . . . . . . 11 ⊢ 0 ≤ ;17
80	59	nn0rei 12514	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℝ
81	80	sqrtsqi 15425	. . . . . . . . . . 11 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
82	79, 81	ax-mp 5	. . . . . . . . . 10 ⊢ (√‘(;17↑2)) = ;17
83	22, 78, 82	3eqtri 2788	. . . . . . . . 9 ⊢ (abs‘(;15 + (i · 8))) = ;17
84	15, 13	crrei 15242	. . . . . . . . 9 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
85	83, 84	oveq12i 7422	. . . . . . . 8 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = (;17 − ;15)
86	1, 2, 28, 24, 36	decaddi 12775	. . . . . . . . 9 ⊢ (;15 + 2) = ;17
87	60, 4, 53, 86	subaddrii 11546	. . . . . . . 8 ⊢ (;17 − ;15) = 2
88	85, 87	eqtri 2784	. . . . . . 7 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = 2
89	88	oveq1i 7420	. . . . . 6 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = (2 / 2)
90		2div2e1 12380	. . . . . 6 ⊢ (2 / 2) = 1
91	89, 90	eqtri 2784	. . . . 5 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = 1
92	91	fveq2i 6884	. . . 4 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘1)
93		sqrt1 15321	. . . 4 ⊢ (√‘1) = 1
94	92, 93	eqtri 2784	. . 3 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = 1
95	20, 94	oveq12i 7422	. 2 ⊢ (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))) = (1 · 1)
96		1t1e1 12401	. 2 ⊢ (1 · 1) = 1
97	10, 95, 96	3eqtri 2788	1 ⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1568 ∈ wcel 2141 ifcif 4486 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 ici 11101 + caddc 11102 · cmul 11104 < clt 11242 ≤ cle 11243 − cmin 11440 -cneg 11441 / cdiv 11870 2c2 12294 4c4 12296 5c5 12297 6c6 12298 7c7 12299 8c8 12300 9c9 12301 cdc 12710 ↑cexp 14096 ℜcre 15147 ℑcim 15148 √csqrt 15283 abscabs 15284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9401 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-rp 13016 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286

This theorem is referenced by: (None)

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