resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 40703

Description: Example for resqrtval 40701. (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 11935	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 11939	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12137	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 11931	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10619	. . . . 5 ⊢ i ∈ ℂ
6		8cn 11756	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10671	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10670	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 40701	. . 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 11941	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 11930	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 11755	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 14686	. . . . . . . 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 13578	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2759	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mulid2i 10669	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 11784	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 11936	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 11931	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 11931	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12199	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 10855	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12183	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulid1i 10668	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7153	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 11815	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2782	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12225	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12188	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12187	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2782	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 13578	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12243	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2782	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7155	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12137	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 11940	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 11938	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2759	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2759	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2759	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 11931	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 11734	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 11818	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 10855	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12182	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 11817	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12176	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12137	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 11931	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 13578	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2759	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 11943	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12137	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mulid2i 10669	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2759	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 11808	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12176	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulid1i 10668	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7153	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12198	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2782	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12236	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12188	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12187	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2783	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2786	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6654	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 11946	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 11930	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 14767	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2786	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 14584	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7153	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 11801	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2782	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12177	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7153	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2759	. . . . . 6 ⊢ ;16 = ;16
83	45	mulid1i 10668	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7153	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2782	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12226	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 10673	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12188	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 11937	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12137	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 11931	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12137	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 11931	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 11763	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11417	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 234	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 11931	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 13578	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12221	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2783	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2786	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6654	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 11946	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 11930	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 14767	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2786	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2112 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 ℂcc 10558 ℝcr 10559 0cc0 10560 1c1 10561 ici 10562 + caddc 10563 · cmul 10565 ≤ cle 10699 / cdiv 11320 2c2 11714 3c3 11715 4c4 11716 5c5 11717 6c6 11718 7c7 11719 8c8 11720 9c9 11721 cdc 12122 ↑cexp 13464 ℜcre 14489 √csqrt 14625 abscabs 14626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-sup 8924 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-rp 12416 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628

This theorem is referenced by: (None)

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