resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 43748

Description: Example for resqrtval 43746. (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 12397	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12401	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12603	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12393	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11065	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12222	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11119	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11118	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 43746	. . 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 12403	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12392	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12221	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15200	. . . . . . . 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 14087	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2731	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11117	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12245	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12398	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12393	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12393	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12665	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11305	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12649	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11116	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7356	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12276	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2754	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12691	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12654	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12653	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2754	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14087	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12709	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2754	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7358	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12603	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12402	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12400	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2731	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2731	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2731	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12393	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12200	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12279	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11305	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12648	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12278	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12642	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12603	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12393	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14087	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2731	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12405	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12603	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11117	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2731	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12269	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12642	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11116	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7356	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12664	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2754	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12702	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12654	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12653	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2755	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2758	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6825	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12408	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12392	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15282	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2758	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15099	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7356	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12262	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2754	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12643	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7356	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2731	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11116	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7356	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2754	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12692	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11121	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12654	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12399	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12603	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12393	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12603	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12393	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12229	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11875	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 231	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12393	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14087	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12687	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2755	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2758	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6825	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12408	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12392	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15282	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2758	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 ici 11008 + caddc 11009 · cmul 11011 ≤ cle 11147 / cdiv 11774 2c2 12180 3c3 12181 4c4 12182 5c5 12183 6c6 12184 7c7 12185 8c8 12186 9c9 12187 cdc 12588 ↑cexp 13968 ℜcre 15004 √csqrt 15140 abscabs 15141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143

This theorem is referenced by: (None)

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