imsqrtvalex - Mathbox for Richard Penner

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Theorem imsqrtvalex 41254

Description: Example for imsqrtval 41252. (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 12249	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12253	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12452	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12245	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10930	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12070	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10982	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10981	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		imsqrtval 41252	. . 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 12085	. . . . 5 ⊢ 0 < 8
12		0re 10977	. . . . . . 7 ⊢ 0 ∈ ℝ
13		8re 12069	. . . . . . 7 ⊢ 8 ∈ ℝ
14	12, 13	ltnsymi 11094	. . . . . 6 ⊢ (0 < 8 → ¬ 8 < 0)
15	3	nn0rei 12244	. . . . . . . 8 ⊢ ;15 ∈ ℝ
16	15, 13	crimi 14904	. . . . . . 7 ⊢ (ℑ‘(;15 + (i · 8))) = 8
17	16	breq1i 5081	. . . . . 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 4469	. . 3 ⊢ if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) = 1
21		absreim 15005	. . . . . . . . . . 11 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
22	15, 13, 21	mp2an 689	. . . . . . . . . 10 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
23	4	sqvali 13897	. . . . . . . . . . . . . 14 ⊢ (;15↑2) = (;15 · ;15)
24		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ;15 = ;15
25		7nn0 12255	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℕ₀
26	4	mulid2i 10980	. . . . . . . . . . . . . . . 16 ⊢ (1 · ;15) = ;15
27		1p1e2 12098	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
28		2nn0 12250	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
29	25	nn0cni 12245	. . . . . . . . . . . . . . . . 17 ⊢ 7 ∈ ℂ
30	2	nn0cni 12245	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
31		7p5e12 12514	. . . . . . . . . . . . . . . . 17 ⊢ (7 + 5) = ;12
32	29, 30, 31	addcomli 11167	. . . . . . . . . . . . . . . 16 ⊢ (5 + 7) = ;12
33	1, 2, 25, 26, 27, 28, 32	decaddci 12498	. . . . . . . . . . . . . . 15 ⊢ ((1 · ;15) + 7) = ;22
34	30	mulid1i 10979	. . . . . . . . . . . . . . . . . 18 ⊢ (5 · 1) = 5
35	34	oveq1i 7285	. . . . . . . . . . . . . . . . 17 ⊢ ((5 · 1) + 2) = (5 + 2)
36		5p2e7 12129	. . . . . . . . . . . . . . . . 17 ⊢ (5 + 2) = 7
37	35, 36	eqtri 2766	. . . . . . . . . . . . . . . 16 ⊢ ((5 · 1) + 2) = 7
38		5t5e25 12540	. . . . . . . . . . . . . . . 16 ⊢ (5 · 5) = ;25
39	2, 1, 2, 24, 2, 28, 37, 38	decmul2c 12503	. . . . . . . . . . . . . . 15 ⊢ (5 · ;15) = ;75
40	3, 1, 2, 24, 2, 25, 33, 39	decmul1c 12502	. . . . . . . . . . . . . 14 ⊢ (;15 · ;15) = ;;225
41	23, 40	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (;15↑2) = ;;225
42	6	sqvali 13897	. . . . . . . . . . . . . 14 ⊢ (8↑2) = (8 · 8)
43		8t8e64 12558	. . . . . . . . . . . . . 14 ⊢ (8 · 8) = ;64
44	42, 43	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (8↑2) = ;64
45	41, 44	oveq12i 7287	. . . . . . . . . . . 12 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
46	28, 28	deccl 12452	. . . . . . . . . . . . 13 ⊢ ;22 ∈ ℕ₀
47		6nn0 12254	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ₀
48		4nn0 12252	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
49		eqid 2738	. . . . . . . . . . . . 13 ⊢ ;;225 = ;;225
50		eqid 2738	. . . . . . . . . . . . 13 ⊢ ;64 = ;64
51		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ;22 = ;22
52	47	nn0cni 12245	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
53	28	nn0cni 12245	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
54		6p2e8 12132	. . . . . . . . . . . . . . 15 ⊢ (6 + 2) = 8
55	52, 53, 54	addcomli 11167	. . . . . . . . . . . . . 14 ⊢ (2 + 6) = 8
56	28, 28, 47, 51, 55	decaddi 12497	. . . . . . . . . . . . 13 ⊢ (;22 + 6) = ;28
57		5p4e9 12131	. . . . . . . . . . . . 13 ⊢ (5 + 4) = 9
58	46, 2, 47, 48, 49, 50, 56, 57	decadd 12491	. . . . . . . . . . . 12 ⊢ (;;225 + ;64) = ;;289
59	1, 25	deccl 12452	. . . . . . . . . . . . . . 15 ⊢ ;17 ∈ ℕ₀
60	59	nn0cni 12245	. . . . . . . . . . . . . 14 ⊢ ;17 ∈ ℂ
61	60	sqvali 13897	. . . . . . . . . . . . 13 ⊢ (;17↑2) = (;17 · ;17)
62		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ;17 = ;17
63		9nn0 12257	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
64	1, 1	deccl 12452	. . . . . . . . . . . . . 14 ⊢ ;11 ∈ ℕ₀
65	60	mulid2i 10980	. . . . . . . . . . . . . . 15 ⊢ (1 · ;17) = ;17
66		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ;11 = ;11
67		7p1e8 12122	. . . . . . . . . . . . . . 15 ⊢ (7 + 1) = 8
68	1, 25, 1, 1, 65, 66, 27, 67	decadd 12491	. . . . . . . . . . . . . 14 ⊢ ((1 · ;17) + ;11) = ;28
69	29	mulid1i 10979	. . . . . . . . . . . . . . . . 17 ⊢ (7 · 1) = 7
70	69	oveq1i 7285	. . . . . . . . . . . . . . . 16 ⊢ ((7 · 1) + 4) = (7 + 4)
71		7p4e11 12513	. . . . . . . . . . . . . . . 16 ⊢ (7 + 4) = ;11
72	70, 71	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ ((7 · 1) + 4) = ;11
73		7t7e49 12551	. . . . . . . . . . . . . . 15 ⊢ (7 · 7) = ;49
74	25, 1, 25, 62, 63, 48, 72, 73	decmul2c 12503	. . . . . . . . . . . . . 14 ⊢ (7 · ;17) = ;;119
75	59, 1, 25, 62, 63, 64, 68, 74	decmul1c 12502	. . . . . . . . . . . . 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 6777	. . . . . . . . . 10 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
79	59	nn0ge0i 12260	. . . . . . . . . . 11 ⊢ 0 ≤ ;17
80	59	nn0rei 12244	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℝ
81	80	sqrtsqi 15086	. . . . . . . . . . 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 14903	. . . . . . . . 9 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
85	83, 84	oveq12i 7287	. . . . . . . 8 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = (;17 − ;15)
86	1, 2, 28, 24, 36	decaddi 12497	. . . . . . . . 9 ⊢ (;15 + 2) = ;17
87	60, 4, 53, 86	subaddrii 11310	. . . . . . . 8 ⊢ (;17 − ;15) = 2
88	85, 87	eqtri 2766	. . . . . . 7 ⊢ ((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) = 2
89	88	oveq1i 7285	. . . . . 6 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = (2 / 2)
90		2div2e1 12114	. . . . . 6 ⊢ (2 / 2) = 1
91	89, 90	eqtri 2766	. . . . 5 ⊢ (((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2) = 1
92	91	fveq2i 6777	. . . 4 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘1)
93		sqrt1 14983	. . . 4 ⊢ (√‘1) = 1
94	92, 93	eqtri 2766	. . 3 ⊢ (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2)) = 1
95	20, 94	oveq12i 7287	. 2 ⊢ (if((ℑ‘(;15 + (i · 8))) < 0, -1, 1) · (√‘(((abs‘(;15 + (i · 8))) − (ℜ‘(;15 + (i · 8)))) / 2))) = (1 · 1)
96		1t1e1 12135	. 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 2106 ifcif 4459 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 4c4 12030 5c5 12031 6c6 12032 7c7 12033 8c8 12034 9c9 12035 cdc 12437 ↑cexp 13782 ℜcre 14808 ℑcim 14809 √csqrt 14944 abscabs 14945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: (None)

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