imsqrtvalex - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > imsqrtvalex		Structured version Visualization version GIF version

Theorem imsqrtvalex 44688

Description: Example for imsqrtval 44686. (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 12622	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12626	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12829	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12618	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11259	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12440	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11316	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11315	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		imsqrtval 44686	. . 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 12458	. . . . 5 ⊢ 0 < 8
12		0re 11310	. . . . . . 7 ⊢ 0 ∈ ℝ
13		8re 12439	. . . . . . 7 ⊢ 8 ∈ ℝ
14	12, 13	ltnsymi 11429	. . . . . 6 ⊢ (0 < 8 → ¬ 8 < 0)
15	3	nn0rei 12617	. . . . . . . 8 ⊢ ;15 ∈ ℝ
16	15, 13	crimi 15360	. . . . . . 7 ⊢ (ℑ‘(;15 + (i · 8))) = 8
17	16	breq1i 5109	. . . . . 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 4491	. . 3 ⊢ if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) = 1
21		absreim 15460	. . . . . . . . . . 11 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
22	15, 13, 21	mp2an 705	. . . . . . . . . 10 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
23	4	sqvali 14323	. . . . . . . . . . . . . 14 ⊢ (;15↑2) = (;15 · ;15)
24		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ;15 = ;15
25		7nn0 12628	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℕ₀
26	4	mullidi 11314	. . . . . . . . . . . . . . . 16 ⊢ (1 · ;15) = ;15
27		1p1e2 12466	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
28		2nn0 12623	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
29	25	nn0cni 12618	. . . . . . . . . . . . . . . . 17 ⊢ 7 ∈ ℂ
30	2	nn0cni 12618	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
31		7p5e12 12896	. . . . . . . . . . . . . . . . 17 ⊢ (7 + 5) = ;12
32	29, 30, 31	addcomli 11502	. . . . . . . . . . . . . . . 16 ⊢ (5 + 7) = ;12
33	1, 2, 25, 26, 27, 28, 32	decaddci 12880	. . . . . . . . . . . . . . 15 ⊢ ((1 · ;15) + 7) = ;22
34	30	mulridi 11313	. . . . . . . . . . . . . . . . . 18 ⊢ (5 · 1) = 5
35	34	oveq1i 7430	. . . . . . . . . . . . . . . . 17 ⊢ ((5 · 1) + 2) = (5 + 2)
36		5p2e7 12498	. . . . . . . . . . . . . . . . 17 ⊢ (5 + 2) = 7
37	35, 36	eqtri 2783	. . . . . . . . . . . . . . . 16 ⊢ ((5 · 1) + 2) = 7
38		5t5e25 12922	. . . . . . . . . . . . . . . 16 ⊢ (5 · 5) = ;25
39	2, 1, 2, 24, 2, 28, 37, 38	decmul2c 12885	. . . . . . . . . . . . . . 15 ⊢ (5 · ;15) = ;75
40	3, 1, 2, 24, 2, 25, 33, 39	decmul1c 12884	. . . . . . . . . . . . . 14 ⊢ (;15 · ;15) = ;;225
41	23, 40	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (;15↑2) = ;;225
42	6	sqvali 14323	. . . . . . . . . . . . . 14 ⊢ (8↑2) = (8 · 8)
43		8t8e64 12940	. . . . . . . . . . . . . 14 ⊢ (8 · 8) = ;64
44	42, 43	eqtri 2783	. . . . . . . . . . . . 13 ⊢ (8↑2) = ;64
45	41, 44	oveq12i 7432	. . . . . . . . . . . 12 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
46	28, 28	deccl 12829	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ₀
47		6nn0 12627	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ₀
48		4nn0 12625	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
49		eqid 2760	. . . . . . . . . . . . 13 ⊢ ;;225 = ;;225
50		eqid 2760	. . . . . . . . . . . . 13 ⊢ ;64 = ;64
51		eqid 2760	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
52	47	nn0cni 12618	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
53	28	nn0cni 12618	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
54		6p2e8 12501	. . . . . . . . . . . . . . 15 ⊢ (6 + 2) = 8
55	52, 53, 54	addcomli 11502	. . . . . . . . . . . . . 14 ⊢ (2 + 6) = 8
56	28, 28, 47, 51, 55	decaddi 12879	. . . . . . . . . . . . 13 ⊢ (;22 + 6) = ;28
57		5p4e9 12500	. . . . . . . . . . . . 13 ⊢ (5 + 4) = 9
58	46, 2, 47, 48, 49, 50, 56, 57	decadd 12873	. . . . . . . . . . . 12 ⊢ (;;225 + ;64) = ;;289
59	1, 25	deccl 12829	. . . . . . . . . . . . . . 15 ⊢ ;17 ∈ ℕ₀
60	59	nn0cni 12618	. . . . . . . . . . . . . 14 ⊢ ;17 ∈ ℂ
61	60	sqvali 14323	. . . . . . . . . . . . 13 ⊢ (;17↑2) = (;17 · ;17)
62		eqid 2760	. . . . . . . . . . . . . 14 ⊢ ;17 = ;17
63		9nn0 12630	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
64	1, 1	deccl 12829	. . . . . . . . . . . . . 14 ⊢ ;11 ∈ ℕ₀
65	60	mullidi 11314	. . . . . . . . . . . . . . 15 ⊢ (1 · ;17) = ;17
66		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ;11 = ;11
67		7p1e8 12491	. . . . . . . . . . . . . . 15 ⊢ (7 + 1) = 8
68	1, 25, 1, 1, 65, 66, 27, 67	decadd 12873	. . . . . . . . . . . . . 14 ⊢ ((1 · ;17) + ;11) = ;28
69	29	mulridi 11313	. . . . . . . . . . . . . . . . 17 ⊢ (7 · 1) = 7
70	69	oveq1i 7430	. . . . . . . . . . . . . . . 16 ⊢ ((7 · 1) + 4) = (7 + 4)
71		7p4e11 12895	. . . . . . . . . . . . . . . 16 ⊢ (7 + 4) = ;11
72	70, 71	eqtri 2783	. . . . . . . . . . . . . . 15 ⊢ ((7 · 1) + 4) = ;11
73		7t7e49 12933	. . . . . . . . . . . . . . 15 ⊢ (7 · 7) = ;49
74	25, 1, 25, 62, 63, 48, 72, 73	decmul2c 12885	. . . . . . . . . . . . . 14 ⊢ (7 · ;17) = ;;119
75	59, 1, 25, 62, 63, 64, 68, 74	decmul1c 12884	. . . . . . . . . . . . 13 ⊢ (;17 · ;17) = ;;289
76	61, 75	eqtr2i 2784	. . . . . . . . . . . 12 ⊢ ;;289 = (;17↑2)
77	45, 58, 76	3eqtri 2787	. . . . . . . . . . 11 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
78	77	fveq2i 6888	. . . . . . . . . 10 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
79	59	nn0ge0i 12633	. . . . . . . . . . 11 ⊢ 0 ≤ ;17
80	59	nn0rei 12617	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℝ
81	80	sqrtsqi 15542	. . . . . . . . . . 11 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
82	79, 81	ax-mp 5	. . . . . . . . . 10 ⊢ (√‘(;17↑2)) = ;17
83	22, 78, 82	3eqtri 2787	. . . . . . . . 9 ⊢ (abs‘(;15 + (i · 8))) = ;17
84	15, 13	crrei 15359	. . . . . . . . 9 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
85	83, 84	oveq12i 7432	. . . . . . . 8 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = (;17 − ;15)
86	1, 2, 28, 24, 36	decaddi 12879	. . . . . . . . 9 ⊢ (;15 + 2) = ;17
87	60, 4, 53, 86	subaddrii 11647	. . . . . . . 8 ⊢ (;17 − ;15) = 2
88	85, 87	eqtri 2783	. . . . . . 7 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = 2
89	88	oveq1i 7430	. . . . . 6 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = (2 / 2)
90		2div2e1 12483	. . . . . 6 ⊢ (2 / 2) = 1
91	89, 90	eqtri 2783	. . . . 5 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = 1
92	91	fveq2i 6888	. . . 4 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘1)
93		sqrt1 15438	. . . 4 ⊢ (√‘1) = 1
94	92, 93	eqtri 2783	. . 3 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = 1
95	20, 94	oveq12i 7432	. 2 ⊢ (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))) = (1 · 1)
96		1t1e1 12504	. 2 ⊢ (1 · 1) = 1
97	10, 95, 96	3eqtri 2787	1 ⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 = wceq 1569 ∈ wcel 2144 ifcif 4481 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 ici 11202 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 − cmin 11541 -cneg 11542 / cdiv 11973 2c2 12397 4c4 12399 5c5 12400 6c6 12401 7c7 12402 8c8 12403 9c9 12404 cdc 12814 ↑cexp 14204 ℜcre 15264 ℑcim 15265 √csqrt 15400 abscabs 15401

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-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

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 4584 df-pr 4586 df-op 4590 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 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9434 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-seq 14145 df-exp 14205 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403

This theorem is used by: (None)

W3C validator