resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 44297

Description: Example for resqrtval 44295. (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 12520	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12524	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12726	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12516	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11159	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12338	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11216	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11215	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 44295	. . 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 12526	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12515	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12337	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15344	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 704	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 14216	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2769	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11214	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12364	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12521	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12516	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12516	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12793	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11402	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12777	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11213	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7421	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12396	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2792	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12819	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12782	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12781	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2792	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14216	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12837	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2792	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7423	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12726	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12525	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12523	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2769	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2769	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2769	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12516	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12316	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12399	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11402	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12776	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12398	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12770	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12726	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12516	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14216	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2769	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12528	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12726	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11214	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2769	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12389	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12770	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11213	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7421	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12792	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2792	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12830	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12782	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12781	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2793	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2796	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6885	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12531	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12515	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15426	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2796	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15243	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7421	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12382	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2792	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12771	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7421	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2769	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11213	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7421	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2792	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12820	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11218	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12782	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12522	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12726	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12516	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12726	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12516	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12347	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11969	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 234	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12516	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14216	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12815	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2793	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2796	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6885	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12531	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12515	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15426	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2796	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1567 ∈ wcel 2149 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 ici 11102 + caddc 11103 · cmul 11105 ≤ cle 11244 / cdiv 11871 2c2 12295 3c3 12296 4c4 12297 5c5 12298 6c6 12299 7c7 12300 8c8 12301 9c9 12302 cdc 12711 ↑cexp 14097 ℜcre 15148 √csqrt 15284 abscabs 15285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9402 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-rp 13017 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287

This theorem is referenced by: (None)

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