resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 44094

Description: Example for resqrtval 44092. (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 12448	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12452	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12654	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12444	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11092	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12273	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11147	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11146	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 44092	. . 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 12454	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12443	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12272	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15250	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 693	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 14137	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2737	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11145	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12296	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12449	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12444	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12444	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12716	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11333	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12700	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11144	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7372	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12327	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2760	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12742	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12705	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12704	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2760	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14137	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12760	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2760	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7374	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12654	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12453	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12451	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2737	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2737	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2737	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12444	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12251	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12330	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11333	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12699	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12329	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12693	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12654	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12444	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14137	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2737	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12456	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12654	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11145	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2737	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12320	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12693	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11144	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7372	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12715	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2760	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12753	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12705	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12704	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2761	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2764	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6839	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12459	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12443	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15332	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2764	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15149	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7372	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12313	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2760	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12694	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7372	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2737	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11144	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7372	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2760	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12743	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11149	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12705	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12450	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12654	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12444	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12654	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12444	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12280	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11904	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 231	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12444	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14137	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12738	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2761	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2764	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6839	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12459	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12443	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15332	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2764	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 class class class wbr 5086 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 ici 11035 + caddc 11036 · cmul 11038 ≤ cle 11175 / cdiv 11802 2c2 12231 3c3 12232 4c4 12233 5c5 12234 6c6 12235 7c7 12236 8c8 12237 9c9 12238 cdc 12639 ↑cexp 14018 ℜcre 15054 √csqrt 15190 abscabs 15191

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-sup 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193

This theorem is referenced by: (None)

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