goldratmolem2 - Mathbox for Ender Ting

	Mathbox for Ender Ting		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > goldratmolem2		Structured version Visualization version GIF version

Theorem goldratmolem2 47349

Description: Lemma 2 for determining the value of golden ratio. (Contributed by Ender Ting, 9-May-2026.)

**Hypothesis**
Ref	Expression
goldra.val	⊢ 𝐹 = (2 · (cos‘(π / 5)))

**Assertion**
Ref	Expression
goldratmolem2	⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

**Proof of Theorem goldratmolem2**
Step	Hyp	Ref	Expression
1		goldra.val	. . 3 ⊢ 𝐹 = (2 · (cos‘(π / 5)))
2	1	goldracos5teq 47348	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26454	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 47344	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11150	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12377	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12448	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14066	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1469	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7367	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14032	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 698	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12247	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14032	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 698	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12276	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12258	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12542	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14047	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1469	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 471	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		1nn0 12444	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
23		6nn0 12449	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
24	22, 23	deccl 12650	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
25	24	nn0cni 12440	. . . . . . . 8 ⊢ ;16 ∈ ℂ
26		6nn 12261	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
27	22, 26	decnncl 12655	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
28	27	nnne0i 12208	. . . . . . . 8 ⊢ ;16 ≠ 0
29	25, 28	pm3.2i 471	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
30		divdiv2 11858	. . . . . . 7 ⊢ (((𝐹↑5) ∈ ℂ ∧ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0) ∧ (;16 ∈ ℂ ∧ ;16 ≠ 0)) → ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5)))
31	12, 21, 29, 30	mp3an 1469	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
32	12, 25	mulcomi 11144	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
33	32	oveq1i 7366	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
34	25, 12, 15, 20	divassi 11902	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
35	31, 33, 34	3eqtrri 2767	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
36		exp1 14020	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
37	13, 36	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
38	37	eqcomi 2748	. . . . . . . . 9 ⊢ 2 = (2↑1)
39		4cn 12257	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
40		ax-1cn 11087	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
41		4p1e5 12313	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
42	39, 40, 41	mvlladdi 11403	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
43	42	oveq2i 7367	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
44	38, 43	eqtri 2762	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
45		4z 12552	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
46	18, 45	pm3.2i 471	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
47		expsub 14063	. . . . . . . . 9 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (5 ∈ ℤ ∧ 4 ∈ ℤ)) → (2↑(5 − 4)) = ((2↑5) / (2↑4)))
48	6, 46, 47	mp2an 698	. . . . . . . 8 ⊢ (2↑(5 − 4)) = ((2↑5) / (2↑4))
49		2exp4 17046	. . . . . . . . 9 ⊢ (2↑4) = ;16
50	49	oveq2i 7367	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
51	44, 48, 50	3eqtri 2766	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
52	51	eqcomi 2748	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
53	52	oveq2i 7367	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
54	10, 35, 53	3eqtri 2766	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
55		3nn0 12446	. . . . . . 7 ⊢ 3 ∈ ℕ₀
56		expdiv 14066	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
57	5, 6, 55, 56	mp3an 1469	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
58	57	oveq2i 7367	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
59		5t4e20 12737	. . . . . . . 8 ⊢ (5 · 4) = ;20
60	59	eqcomi 2748	. . . . . . 7 ⊢ ;20 = (5 · 4)
61	60	oveq1i 7366	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
62		5cn 12260	. . . . . . 7 ⊢ 5 ∈ ℂ
63		expcl 14032	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
64	5, 55, 63	mp2an 698	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
65		expcl 14032	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
66	13, 55, 65	mp2an 698	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
67		3z 12551	. . . . . . . . 9 ⊢ 3 ∈ ℤ
68		expne0i 14047	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
69	13, 16, 67, 68	mp3an 1469	. . . . . . . 8 ⊢ (2↑3) ≠ 0
70	64, 66, 69	divcli 11888	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
71	62, 39, 70	mulassi 11147	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
72	61, 71	eqtri 2762	. . . . 5 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
73	66, 69	pm3.2i 471	. . . . . . . . 9 ⊢ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0)
74		4ne0 12280	. . . . . . . . . 10 ⊢ 4 ≠ 0
75	39, 74	pm3.2i 471	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
76		divdiv2 11858	. . . . . . . . 9 ⊢ (((𝐹↑3) ∈ ℂ ∧ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3)))
77	64, 73, 75, 76	mp3an 1469	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
78	64, 39	mulcomi 11144	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
79	78	oveq1i 7366	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
80	39, 64, 66, 69	divassi 11902	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
81	77, 79, 80	3eqtrri 2767	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
82		4t2e8 12335	. . . . . . . . . 10 ⊢ (4 · 2) = 8
83		cu2 14153	. . . . . . . . . . 11 ⊢ (2↑3) = 8
84	83	eqcomi 2748	. . . . . . . . . 10 ⊢ 8 = (2↑3)
85	82, 84	eqtri 2762	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
86	66, 39, 13, 74	divmuli 11900	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
87	85, 86	mpbir 232	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
88	87	oveq2i 7367	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
89	81, 88	eqtri 2762	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
90	89	oveq2i 7367	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
91	58, 72, 90	3eqtri 2766	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
92	54, 91	oveq12i 7368	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
93	92	oveq1i 7366	. 2 ⊢ (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2))) = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))
94	2, 3, 93	3eqtr3i 2770	1 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 3c3 12228 4c4 12229 5c5 12230 6c6 12231 8c8 12233 ℕ₀cn0 12428 ℤcz 12515 cdc 12635 ↑cexp 14014 cosccos 16020 πcpi 16022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852

This theorem is referenced by: (None)

W3C validator