resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 41184

Description: Example for resqrtval 41182. (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 12195	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12199	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12397	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12191	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10877	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12016	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10929	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10928	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 41182	. . 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 12201	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12190	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12015	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 14949	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 688	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 13841	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2737	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mulid2i 10927	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12044	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12196	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12191	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12191	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12459	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11113	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12443	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulid1i 10926	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7270	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12075	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2765	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12485	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12448	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12447	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2765	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 13841	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12503	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2765	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7272	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12397	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12200	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12198	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2737	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2737	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2737	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12191	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 11994	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12078	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11113	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12442	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12077	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12436	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12397	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12191	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 13841	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2737	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12203	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12397	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mulid2i 10927	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2737	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12068	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12436	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulid1i 10926	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7270	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12458	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2765	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12496	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12448	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12447	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2766	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2769	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6764	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12206	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12190	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15030	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2769	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 14847	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7270	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12061	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2765	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12437	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7270	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2737	. . . . . 6 ⊢ ;16 = ;16
83	45	mulid1i 10926	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7270	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2765	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12486	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 10931	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12448	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12197	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12397	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12191	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12397	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12191	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12023	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11675	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 230	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12191	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 13841	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12481	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2766	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2769	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6764	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12206	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12190	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15030	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2769	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2107 class class class wbr 5075 ‘cfv 6423 (class class class)co 7260 ℂcc 10816 ℝcr 10817 0cc0 10818 1c1 10819 ici 10820 + caddc 10821 · cmul 10823 ≤ cle 10957 / cdiv 11578 2c2 11974 3c3 11975 4c4 11976 5c5 11977 6c6 11978 7c7 11979 8c8 11980 9c9 11981 cdc 12382 ↑cexp 13726 ℜcre 14752 √csqrt 14888 abscabs 14889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 ax-pre-sup 10896

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-sup 9147 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-3 11983 df-4 11984 df-5 11985 df-6 11986 df-7 11987 df-8 11988 df-9 11989 df-n0 12180 df-z 12266 df-dec 12383 df-uz 12528 df-rp 12676 df-seq 13666 df-exp 13727 df-cj 14754 df-re 14755 df-im 14756 df-sqrt 14890 df-abs 14891

This theorem is referenced by: (None)

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