resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 40345

Description: Example for resqrtval 40343. (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 11901	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 11905	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12101	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 11897	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10585	. . . . 5 ⊢ i ∈ ℂ
6		8cn 11722	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10637	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10636	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 40343	. . 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 11907	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 11896	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 11721	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 14645	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 691	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 13539	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2798	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mulid2i 10635	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 11750	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 11902	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 11897	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 11897	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12163	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 10821	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12147	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulid1i 10634	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7145	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 11781	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2821	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12189	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12152	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12151	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2821	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 13539	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12207	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2821	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7147	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12101	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 11906	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 11904	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2798	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2798	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2798	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 11897	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 11700	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 11784	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 10821	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12146	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 11783	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12140	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12101	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 11897	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 13539	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2798	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 11909	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12101	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mulid2i 10635	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2798	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 11774	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12140	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulid1i 10634	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7145	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12162	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2821	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12200	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12152	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12151	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2822	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2825	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6648	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 11912	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 11896	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 14726	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2825	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 14543	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7145	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 11767	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2821	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12141	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7145	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2798	. . . . . 6 ⊢ ;16 = ;16
83	45	mulid1i 10634	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7145	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2821	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12190	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 10639	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12152	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 11903	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12101	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 11897	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12101	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 11897	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 11729	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11383	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 234	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 11897	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 13539	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12185	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2822	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2825	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6648	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 11912	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 11896	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 14726	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2825	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 ≤ cle 10665 / cdiv 11286 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 7c7 11685 8c8 11686 9c9 11687 cdc 12086 ↑cexp 13425 ℜcre 14448 √csqrt 14584 abscabs 14585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587

This theorem is referenced by: (None)

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