goldratmolem2 - Mathbox for Ender Ting

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Theorem goldratmolem2 47361

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 47360	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26457	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 47356	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11155	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12381	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12452	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14070	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1470	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7370	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14036	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 699	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12251	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14036	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 699	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12280	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12262	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12546	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14051	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1470	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 472	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		1nn0 12448	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
23		6nn0 12453	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
24	22, 23	deccl 12654	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
25	24	nn0cni 12444	. . . . . . . 8 ⊢ ;16 ∈ ℂ
26		6nn 12265	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
27	22, 26	decnncl 12659	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
28	27	nnne0i 12212	. . . . . . . 8 ⊢ ;16 ≠ 0
29	25, 28	pm3.2i 472	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
30		divdiv2 11862	. . . . . . 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 1470	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
32	12, 25	mulcomi 11149	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
33	32	oveq1i 7369	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
34	25, 12, 15, 20	divassi 11906	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
35	31, 33, 34	3eqtrri 2769	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
36		exp1 14024	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
37	13, 36	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
38	37	eqcomi 2750	. . . . . . . . 9 ⊢ 2 = (2↑1)
39		4cn 12261	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
40		ax-1cn 11092	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
41		4p1e5 12317	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
42	39, 40, 41	mvlladdi 11408	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
43	42	oveq2i 7370	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
44	38, 43	eqtri 2764	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
45		4z 12556	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
46	18, 45	pm3.2i 472	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
47		expsub 14067	. . . . . . . . 9 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (5 ∈ ℤ ∧ 4 ∈ ℤ)) → (2↑(5 − 4)) = ((2↑5) / (2↑4)))
48	6, 46, 47	mp2an 699	. . . . . . . 8 ⊢ (2↑(5 − 4)) = ((2↑5) / (2↑4))
49		2exp4 17050	. . . . . . . . 9 ⊢ (2↑4) = ;16
50	49	oveq2i 7370	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
51	44, 48, 50	3eqtri 2768	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
52	51	eqcomi 2750	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
53	52	oveq2i 7370	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
54	10, 35, 53	3eqtri 2768	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
55		3nn0 12450	. . . . . . 7 ⊢ 3 ∈ ℕ₀
56		expdiv 14070	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
57	5, 6, 55, 56	mp3an 1470	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
58	57	oveq2i 7370	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
59		5t4e20 12741	. . . . . . . 8 ⊢ (5 · 4) = ;20
60	59	eqcomi 2750	. . . . . . 7 ⊢ ;20 = (5 · 4)
61	60	oveq1i 7369	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
62		5cn 12264	. . . . . . 7 ⊢ 5 ∈ ℂ
63		expcl 14036	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
64	5, 55, 63	mp2an 699	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
65		expcl 14036	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
66	13, 55, 65	mp2an 699	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
67		3z 12555	. . . . . . . . 9 ⊢ 3 ∈ ℤ
68		expne0i 14051	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
69	13, 16, 67, 68	mp3an 1470	. . . . . . . 8 ⊢ (2↑3) ≠ 0
70	64, 66, 69	divcli 11892	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
71	62, 39, 70	mulassi 11152	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
72	61, 71	eqtri 2764	. . . . 5 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
73	66, 69	pm3.2i 472	. . . . . . . . 9 ⊢ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0)
74		4ne0 12284	. . . . . . . . . 10 ⊢ 4 ≠ 0
75	39, 74	pm3.2i 472	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
76		divdiv2 11862	. . . . . . . . 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 1470	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
78	64, 39	mulcomi 11149	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
79	78	oveq1i 7369	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
80	39, 64, 66, 69	divassi 11906	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
81	77, 79, 80	3eqtrri 2769	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
82		4t2e8 12339	. . . . . . . . . 10 ⊢ (4 · 2) = 8
83		cu2 14157	. . . . . . . . . . 11 ⊢ (2↑3) = 8
84	83	eqcomi 2750	. . . . . . . . . 10 ⊢ 8 = (2↑3)
85	82, 84	eqtri 2764	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
86	66, 39, 13, 74	divmuli 11904	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
87	85, 86	mpbir 233	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
88	87	oveq2i 7370	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
89	81, 88	eqtri 2764	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
90	89	oveq2i 7370	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
91	58, 72, 90	3eqtri 2768	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
92	54, 91	oveq12i 7371	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
93	92	oveq1i 7369	. 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 2772	1 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ‘cfv 6488 (class class class)co 7359 ℂcc 11032 0cc0 11034 1c1 11035 + caddc 11037 · cmul 11039 − cmin 11373 -cneg 11374 / cdiv 11803 2c2 12231 3c3 12232 4c4 12233 5c5 12234 6c6 12235 8c8 12237 ℕ₀cn0 12432 ℤcz 12519 cdc 12639 ↑cexp 14018 cosccos 16024 πcpi 16026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 ax-addf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19286 df-cmn 19751 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-fbas 21347 df-fg 21348 df-cnfld 21351 df-top 22880 df-topon 22897 df-topsp 22919 df-bases 22932 df-cld 23005 df-ntr 23006 df-cls 23007 df-nei 23084 df-lp 23122 df-perf 23123 df-cn 23213 df-cnp 23214 df-haus 23301 df-tx 23548 df-hmeo 23741 df-fil 23832 df-fm 23924 df-flim 23925 df-flf 23926 df-xms 24306 df-ms 24307 df-tms 24308 df-cncf 24866 df-limc 25854 df-dv 25855

This theorem is referenced by: (None)

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