resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 43663

Description: Example for resqrtval 43661. (Contributed by RP, 21-May-2024.)

**Assertion**
Ref	Expression
resqrtvalex	⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

**Proof of Theorem resqrtvalex**
Step	Hyp	Ref	Expression
1		1nn0 12544	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12548	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12750	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12540	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11215	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12364	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11269	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11268	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 43661	. . 3 ⊢ ((;15 + (i · 8)) ∈ ℂ → (ℜ‘(√‘(;15 + (i · 8)))) = (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)))
10	8, 9	ax-mp 5	. 2 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2))
11		7nn0 12550	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12539	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12363	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15333	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 692	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 14220	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2736	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11267	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12392	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12545	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12540	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12540	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12812	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11454	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12796	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11266	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7442	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12423	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2764	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12838	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12801	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12800	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2764	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14220	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12856	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2764	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7444	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12750	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12549	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12547	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2736	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2736	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2736	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12540	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12342	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12426	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11454	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12795	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12425	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12789	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12750	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12540	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14220	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2736	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12552	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12750	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11267	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2736	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12416	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12789	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11266	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7442	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12811	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2764	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12849	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12801	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12800	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2765	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2768	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6908	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12555	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12539	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15414	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2768	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15232	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7442	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12409	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2764	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12790	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7442	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2736	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11266	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7442	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2764	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12839	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11271	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12801	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12546	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12750	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12540	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12750	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12540	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12371	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 12022	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 231	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12540	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14220	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12834	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2765	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2768	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6908	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12555	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12539	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15414	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2768	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 ici 11158 + caddc 11159 · cmul 11161 ≤ cle 11297 / cdiv 11921 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 cdc 12735 ↑cexp 14103 ℜcre 15137 √csqrt 15273 abscabs 15274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276

This theorem is referenced by: (None)

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