resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 41206

Description: Example for resqrtval 41204. (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 12232	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12236	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12434	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12228	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 10914	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12053	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 10966	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 10965	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 41204	. . 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 12238	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12227	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12052	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 14986	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 688	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 13878	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2739	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mulid2i 10964	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12081	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12233	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12228	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12228	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12496	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11150	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12480	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulid1i 10963	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7278	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12112	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2767	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12522	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12485	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12484	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2767	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 13878	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12540	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2767	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7280	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12434	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12237	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12235	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2739	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2739	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2739	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12228	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12031	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12115	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11150	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12479	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12114	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12473	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12434	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12228	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 13878	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2739	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12240	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12434	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mulid2i 10964	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2739	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12105	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12473	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulid1i 10963	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7278	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12495	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2767	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12533	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12485	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12484	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2768	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2771	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6771	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12243	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12227	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15067	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2771	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 14884	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7278	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12098	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2767	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12474	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7278	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2739	. . . . . 6 ⊢ ;16 = ;16
83	45	mulid1i 10963	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7278	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2767	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12523	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 10968	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12485	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12234	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12434	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12228	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12434	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12228	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12060	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11712	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 230	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12228	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 13878	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12518	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2768	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2771	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6771	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12243	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12227	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15067	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2771	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2109 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 0cc0 10855 1c1 10856 ici 10857 + caddc 10858 · cmul 10860 ≤ cle 10994 / cdiv 11615 2c2 12011 3c3 12012 4c4 12013 5c5 12014 6c6 12015 7c7 12016 8c8 12017 9c9 12018 cdc 12419 ↑cexp 13763 ℜcre 14789 √csqrt 14925 abscabs 14926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-rp 12713 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928

This theorem is referenced by: (None)

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