resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 42382

Description: Example for resqrtval 42380. (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 12485	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12489	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12689	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12481	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11166	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12306	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11218	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11217	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 42380	. . 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 12491	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12480	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12305	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15237	. . . . . . . 8 ⊢ ((;15 ∈ ℝ ∧ 8 ∈ ℝ) → (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2))))
15	12, 13, 14	mp2an 691	. . . . . . 7 ⊢ (abs‘(;15 + (i · 8))) = (√‘((;15↑2) + (8↑2)))
16	4	sqvali 14141	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2733	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11216	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12334	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12486	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12481	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12481	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12751	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11403	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12735	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11215	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7416	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12365	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2761	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12777	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12740	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12739	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2761	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14141	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12795	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2761	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7418	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12689	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12490	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12488	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2733	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2733	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2733	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12481	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12284	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12368	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11403	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12734	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12367	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12728	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12689	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12481	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14141	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2733	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12493	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12689	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11216	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2733	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12358	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12728	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11215	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7416	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12750	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2761	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12788	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12740	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12739	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2762	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2765	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6892	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12496	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12480	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15318	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2765	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15136	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7416	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12351	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2761	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12729	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7416	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2733	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11215	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7416	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2761	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12778	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11220	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12740	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12487	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12689	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12481	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12689	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12481	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12313	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11965	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 230	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12481	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14141	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12773	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2762	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2765	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6892	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12496	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12480	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15318	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2765	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 ici 11109 + caddc 11110 · cmul 11112 ≤ cle 11246 / cdiv 11868 2c2 12264 3c3 12265 4c4 12266 5c5 12267 6c6 12268 7c7 12269 8c8 12270 9c9 12271 cdc 12674 ↑cexp 14024 ℜcre 15041 √csqrt 15177 abscabs 15178

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180

This theorem is referenced by: (None)

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