resqrtvalex - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > resqrtvalex		Structured version Visualization version GIF version

Theorem resqrtvalex 44226

Description: Example for resqrtval 44224. (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 12499	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12503	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12705	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12495	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11134	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12317	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11191	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11190	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 44224	. . 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 12505	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12494	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12316	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15322	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 702	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 14195	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2764	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11189	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12343	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12500	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12495	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12495	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12772	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11377	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12756	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11188	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7408	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12375	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2787	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12798	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12761	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12760	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2787	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14195	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12816	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2787	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7410	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12705	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12504	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12502	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2764	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2764	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2764	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12495	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12295	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12378	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11377	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12755	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12377	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12749	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12705	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12495	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14195	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2764	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12507	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12705	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11189	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2764	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12368	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12749	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11188	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7408	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12771	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2787	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12809	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12761	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12760	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2788	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2791	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6872	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12510	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12494	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15404	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2791	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15221	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7408	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12361	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2787	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12750	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7408	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2764	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11188	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7408	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2787	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12799	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11193	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12761	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12501	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12705	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12495	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12705	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12495	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12326	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11947	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 233	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12495	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14195	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12794	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2788	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2791	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6872	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12510	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12494	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15404	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2791	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1562 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 ici 11077 + caddc 11078 · cmul 11080 ≤ cle 11219 / cdiv 11846 2c2 12274 3c3 12275 4c4 12276 5c5 12277 6c6 12278 7c7 12279 8c8 12280 9c9 12281 cdc 12690 ↑cexp 14076 ℜcre 15126 √csqrt 15262 abscabs 15263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-sup 9390 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-rp 12996 df-seq 14017 df-exp 14077 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265

This theorem is referenced by: (None)

W3C validator