resqrtvalex - Mathbox for Richard Penner

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Theorem resqrtvalex 43627

Description: Example for resqrtval 43625. (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 12434	. . . . . 6 ⊢ 1 ∈ ℕ₀
2		5nn0 12438	. . . . . 6 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12640	. . . . 5 ⊢ ;15 ∈ ℕ₀
4	3	nn0cni 12430	. . . 4 ⊢ ;15 ∈ ℂ
5		ax-icn 11103	. . . . 5 ⊢ i ∈ ℂ
6		8cn 12259	. . . . 5 ⊢ 8 ∈ ℂ
7	5, 6	mulcli 11157	. . . 4 ⊢ (i · 8) ∈ ℂ
8	4, 7	addcli 11156	. . 3 ⊢ (;15 + (i · 8)) ∈ ℂ
9		resqrtval 43625	. . 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 12440	. . . . . 6 ⊢ 7 ∈ ℕ₀
12	3	nn0rei 12429	. . . . . . . 8 ⊢ ;15 ∈ ℝ
13		8re 12258	. . . . . . . 8 ⊢ 8 ∈ ℝ
14		absreim 15235	. . . . . . . 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 14121	. . . . . . . . . . 11 ⊢ (;15↑2) = (;15 · ;15)
17		eqid 2729	. . . . . . . . . . . 12 ⊢ ;15 = ;15
18	4	mullidi 11155	. . . . . . . . . . . . 13 ⊢ (1 · ;15) = ;15
19		1p1e2 12282	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
20		2nn0 12435	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
21	11	nn0cni 12430	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
22	2	nn0cni 12430	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℂ
23		7p5e12 12702	. . . . . . . . . . . . . 14 ⊢ (7 + 5) = ;12
24	21, 22, 23	addcomli 11342	. . . . . . . . . . . . 13 ⊢ (5 + 7) = ;12
25	1, 2, 11, 18, 19, 20, 24	decaddci 12686	. . . . . . . . . . . 12 ⊢ ((1 · ;15) + 7) = ;22
26	22	mulridi 11154	. . . . . . . . . . . . . . 15 ⊢ (5 · 1) = 5
27	26	oveq1i 7379	. . . . . . . . . . . . . 14 ⊢ ((5 · 1) + 2) = (5 + 2)
28		5p2e7 12313	. . . . . . . . . . . . . 14 ⊢ (5 + 2) = 7
29	27, 28	eqtri 2752	. . . . . . . . . . . . 13 ⊢ ((5 · 1) + 2) = 7
30		5t5e25 12728	. . . . . . . . . . . . 13 ⊢ (5 · 5) = ;25
31	2, 1, 2, 17, 2, 20, 29, 30	decmul2c 12691	. . . . . . . . . . . 12 ⊢ (5 · ;15) = ;75
32	3, 1, 2, 17, 2, 11, 25, 31	decmul1c 12690	. . . . . . . . . . 11 ⊢ (;15 · ;15) = ;;225
33	16, 32	eqtri 2752	. . . . . . . . . 10 ⊢ (;15↑2) = ;;225
34	6	sqvali 14121	. . . . . . . . . . 11 ⊢ (8↑2) = (8 · 8)
35		8t8e64 12746	. . . . . . . . . . 11 ⊢ (8 · 8) = ;64
36	34, 35	eqtri 2752	. . . . . . . . . 10 ⊢ (8↑2) = ;64
37	33, 36	oveq12i 7381	. . . . . . . . 9 ⊢ ((;15↑2) + (8↑2)) = (;;225 + ;64)
38	20, 20	deccl 12640	. . . . . . . . . 10 ⊢ ;22 ∈ ℕ₀
39		6nn0 12439	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
40		4nn0 12437	. . . . . . . . . 10 ⊢ 4 ∈ ℕ₀
41		eqid 2729	. . . . . . . . . 10 ⊢ ;;225 = ;;225
42		eqid 2729	. . . . . . . . . 10 ⊢ ;64 = ;64
43		eqid 2729	. . . . . . . . . . 11 ⊢ ;22 = ;22
44	39	nn0cni 12430	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
45		2cn 12237	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
46		6p2e8 12316	. . . . . . . . . . . 12 ⊢ (6 + 2) = 8
47	44, 45, 46	addcomli 11342	. . . . . . . . . . 11 ⊢ (2 + 6) = 8
48	20, 20, 39, 43, 47	decaddi 12685	. . . . . . . . . 10 ⊢ (;22 + 6) = ;28
49		5p4e9 12315	. . . . . . . . . 10 ⊢ (5 + 4) = 9
50	38, 2, 39, 40, 41, 42, 48, 49	decadd 12679	. . . . . . . . 9 ⊢ (;;225 + ;64) = ;;289
51	1, 11	deccl 12640	. . . . . . . . . . . 12 ⊢ ;17 ∈ ℕ₀
52	51	nn0cni 12430	. . . . . . . . . . 11 ⊢ ;17 ∈ ℂ
53	52	sqvali 14121	. . . . . . . . . 10 ⊢ (;17↑2) = (;17 · ;17)
54		eqid 2729	. . . . . . . . . . 11 ⊢ ;17 = ;17
55		9nn0 12442	. . . . . . . . . . 11 ⊢ 9 ∈ ℕ₀
56	1, 1	deccl 12640	. . . . . . . . . . 11 ⊢ ;11 ∈ ℕ₀
57	52	mullidi 11155	. . . . . . . . . . . 12 ⊢ (1 · ;17) = ;17
58		eqid 2729	. . . . . . . . . . . 12 ⊢ ;11 = ;11
59		7p1e8 12306	. . . . . . . . . . . 12 ⊢ (7 + 1) = 8
60	1, 11, 1, 1, 57, 58, 19, 59	decadd 12679	. . . . . . . . . . 11 ⊢ ((1 · ;17) + ;11) = ;28
61	21	mulridi 11154	. . . . . . . . . . . . . 14 ⊢ (7 · 1) = 7
62	61	oveq1i 7379	. . . . . . . . . . . . 13 ⊢ ((7 · 1) + 4) = (7 + 4)
63		7p4e11 12701	. . . . . . . . . . . . 13 ⊢ (7 + 4) = ;11
64	62, 63	eqtri 2752	. . . . . . . . . . . 12 ⊢ ((7 · 1) + 4) = ;11
65		7t7e49 12739	. . . . . . . . . . . 12 ⊢ (7 · 7) = ;49
66	11, 1, 11, 54, 55, 40, 64, 65	decmul2c 12691	. . . . . . . . . . 11 ⊢ (7 · ;17) = ;;119
67	51, 1, 11, 54, 55, 56, 60, 66	decmul1c 12690	. . . . . . . . . 10 ⊢ (;17 · ;17) = ;;289
68	53, 67	eqtr2i 2753	. . . . . . . . 9 ⊢ ;;289 = (;17↑2)
69	37, 50, 68	3eqtri 2756	. . . . . . . 8 ⊢ ((;15↑2) + (8↑2)) = (;17↑2)
70	69	fveq2i 6843	. . . . . . 7 ⊢ (√‘((;15↑2) + (8↑2))) = (√‘(;17↑2))
71	51	nn0ge0i 12445	. . . . . . . 8 ⊢ 0 ≤ ;17
72	51	nn0rei 12429	. . . . . . . . 9 ⊢ ;17 ∈ ℝ
73	72	sqrtsqi 15317	. . . . . . . 8 ⊢ (0 ≤ ;17 → (√‘(;17↑2)) = ;17)
74	71, 73	ax-mp 5	. . . . . . 7 ⊢ (√‘(;17↑2)) = ;17
75	15, 70, 74	3eqtri 2756	. . . . . 6 ⊢ (abs‘(;15 + (i · 8))) = ;17
76	12, 13	crrei 15134	. . . . . 6 ⊢ (ℜ‘(;15 + (i · 8))) = ;15
77	19	oveq1i 7379	. . . . . . 7 ⊢ ((1 + 1) + 1) = (2 + 1)
78		2p1e3 12299	. . . . . . 7 ⊢ (2 + 1) = 3
79	77, 78	eqtri 2752	. . . . . 6 ⊢ ((1 + 1) + 1) = 3
80	1, 11, 1, 2, 75, 76, 79, 20, 23	decaddc 12680	. . . . 5 ⊢ ((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) = ;32
81	80	oveq1i 7379	. . . 4 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (;32 / 2)
82		eqid 2729	. . . . . 6 ⊢ ;16 = ;16
83	45	mulridi 11154	. . . . . . . 8 ⊢ (2 · 1) = 2
84	83	oveq1i 7379	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
85	84, 78	eqtri 2752	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
86		6t2e12 12729	. . . . . . 7 ⊢ (6 · 2) = ;12
87	44, 45, 86	mulcomli 11159	. . . . . 6 ⊢ (2 · 6) = ;12
88	20, 1, 39, 82, 20, 1, 85, 87	decmul2c 12691	. . . . 5 ⊢ (2 · ;16) = ;32
89		3nn0 12436	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
90	89, 20	deccl 12640	. . . . . . 7 ⊢ ;32 ∈ ℕ₀
91	90	nn0cni 12430	. . . . . 6 ⊢ ;32 ∈ ℂ
92	1, 39	deccl 12640	. . . . . . 7 ⊢ ;16 ∈ ℕ₀
93	92	nn0cni 12430	. . . . . 6 ⊢ ;16 ∈ ℂ
94		2ne0 12266	. . . . . 6 ⊢ 2 ≠ 0
95	91, 45, 93, 94	divmuli 11912	. . . . 5 ⊢ ((;32 / 2) = ;16 ↔ (2 · ;16) = ;32)
96	88, 95	mpbir 231	. . . 4 ⊢ (;32 / 2) = ;16
97	40	nn0cni 12430	. . . . . 6 ⊢ 4 ∈ ℂ
98	97	sqvali 14121	. . . . 5 ⊢ (4↑2) = (4 · 4)
99		4t4e16 12724	. . . . 5 ⊢ (4 · 4) = ;16
100	98, 99	eqtr2i 2753	. . . 4 ⊢ ;16 = (4↑2)
101	81, 96, 100	3eqtri 2756	. . 3 ⊢ (((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2) = (4↑2)
102	101	fveq2i 6843	. 2 ⊢ (√‘(((abs‘(;15 + (i · 8))) + (ℜ‘(;15 + (i · 8)))) / 2)) = (√‘(4↑2))
103	40	nn0ge0i 12445	. . 3 ⊢ 0 ≤ 4
104	40	nn0rei 12429	. . . 4 ⊢ 4 ∈ ℝ
105	104	sqrtsqi 15317	. . 3 ⊢ (0 ≤ 4 → (√‘(4↑2)) = 4)
106	103, 105	ax-mp 5	. 2 ⊢ (√‘(4↑2)) = 4
107	10, 102, 106	3eqtri 2756	1 ⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 ici 11046 + caddc 11047 · cmul 11049 ≤ cle 11185 / cdiv 11811 2c2 12217 3c3 12218 4c4 12219 5c5 12220 6c6 12221 7c7 12222 8c8 12223 9c9 12224 cdc 12625 ↑cexp 14002 ℜcre 15039 √csqrt 15175 abscabs 15176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178

This theorem is referenced by: (None)

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