goldratmolem2 - Mathbox for Ender Ting

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Theorem goldratmolem2 47334

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 47333	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26436	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 47329	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11159	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12386	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12457	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14075	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1464	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7378	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14041	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 693	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12256	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14041	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 693	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12285	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12267	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12551	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14056	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1464	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 470	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		1nn0 12453	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
23		6nn0 12458	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
24	22, 23	deccl 12659	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
25	24	nn0cni 12449	. . . . . . . 8 ⊢ ;16 ∈ ℂ
26		6nn 12270	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
27	22, 26	decnncl 12664	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
28	27	nnne0i 12217	. . . . . . . 8 ⊢ ;16 ≠ 0
29	25, 28	pm3.2i 470	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
30		divdiv2 11867	. . . . . . 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 1464	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
32	12, 25	mulcomi 11153	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
33	32	oveq1i 7377	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
34	25, 12, 15, 20	divassi 11911	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
35	31, 33, 34	3eqtrri 2764	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
36		exp1 14029	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
37	13, 36	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
38	37	eqcomi 2745	. . . . . . . . 9 ⊢ 2 = (2↑1)
39		4cn 12266	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
40		ax-1cn 11096	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
41		4p1e5 12322	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
42	39, 40, 41	mvlladdi 11412	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
43	42	oveq2i 7378	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
44	38, 43	eqtri 2759	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
45		4z 12561	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
46	18, 45	pm3.2i 470	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
47		expsub 14072	. . . . . . . . 9 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (5 ∈ ℤ ∧ 4 ∈ ℤ)) → (2↑(5 − 4)) = ((2↑5) / (2↑4)))
48	6, 46, 47	mp2an 693	. . . . . . . 8 ⊢ (2↑(5 − 4)) = ((2↑5) / (2↑4))
49		2exp4 17055	. . . . . . . . 9 ⊢ (2↑4) = ;16
50	49	oveq2i 7378	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
51	44, 48, 50	3eqtri 2763	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
52	51	eqcomi 2745	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
53	52	oveq2i 7378	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
54	10, 35, 53	3eqtri 2763	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
55		3nn0 12455	. . . . . . 7 ⊢ 3 ∈ ℕ₀
56		expdiv 14075	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
57	5, 6, 55, 56	mp3an 1464	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
58	57	oveq2i 7378	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
59		5t4e20 12746	. . . . . . . 8 ⊢ (5 · 4) = ;20
60	59	eqcomi 2745	. . . . . . 7 ⊢ ;20 = (5 · 4)
61	60	oveq1i 7377	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
62		5cn 12269	. . . . . . 7 ⊢ 5 ∈ ℂ
63		expcl 14041	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
64	5, 55, 63	mp2an 693	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
65		expcl 14041	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
66	13, 55, 65	mp2an 693	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
67		3z 12560	. . . . . . . . 9 ⊢ 3 ∈ ℤ
68		expne0i 14056	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
69	13, 16, 67, 68	mp3an 1464	. . . . . . . 8 ⊢ (2↑3) ≠ 0
70	64, 66, 69	divcli 11897	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
71	62, 39, 70	mulassi 11156	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
72	61, 71	eqtri 2759	. . . . 5 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
73	66, 69	pm3.2i 470	. . . . . . . . 9 ⊢ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0)
74		4ne0 12289	. . . . . . . . . 10 ⊢ 4 ≠ 0
75	39, 74	pm3.2i 470	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
76		divdiv2 11867	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
78	64, 39	mulcomi 11153	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
79	78	oveq1i 7377	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
80	39, 64, 66, 69	divassi 11911	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
81	77, 79, 80	3eqtrri 2764	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
82		4t2e8 12344	. . . . . . . . . 10 ⊢ (4 · 2) = 8
83		cu2 14162	. . . . . . . . . . 11 ⊢ (2↑3) = 8
84	83	eqcomi 2745	. . . . . . . . . 10 ⊢ 8 = (2↑3)
85	82, 84	eqtri 2759	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
86	66, 39, 13, 74	divmuli 11909	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
87	85, 86	mpbir 231	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
88	87	oveq2i 7378	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
89	81, 88	eqtri 2759	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
90	89	oveq2i 7378	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
91	58, 72, 90	3eqtri 2763	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
92	54, 91	oveq12i 7379	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
93	92	oveq1i 7377	. 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 2767	1 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11377 -cneg 11378 / cdiv 11807 2c2 12236 3c3 12237 4c4 12238 5c5 12239 6c6 12240 8c8 12242 ℕ₀cn0 12437 ℤcz 12524 cdc 12644 ↑cexp 14023 cosccos 16029 πcpi 16031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834

This theorem is referenced by: (None)

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