goldratmolem2 - Mathbox for Ender Ting

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Theorem goldratmolem2 47568

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 47567	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26613	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 47563	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11222	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12452	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12523	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14148	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1488	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7421	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14114	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 704	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12315	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14114	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 704	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12346	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12326	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12617	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14129	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1488	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 475	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		1nn0 12519	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
23		6nn0 12524	. . . . . . . . . 10 ⊢ 6 ∈ ℕ₀
24	22, 23	deccl 12725	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
25	24	nn0cni 12515	. . . . . . . 8 ⊢ ;16 ∈ ℂ
26		6nn 12329	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
27	22, 26	decnncl 12734	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
28	27	nnne0i 12275	. . . . . . . 8 ⊢ ;16 ≠ 0
29	25, 28	pm3.2i 475	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
30		divdiv2 11926	. . . . . . 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 1488	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
32	12, 25	mulcomi 11216	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
33	32	oveq1i 7420	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
34	25, 12, 15, 20	divassi 11970	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
35	31, 33, 34	3eqtrri 2789	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
36		exp1 14102	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
37	13, 36	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
38	37	eqcomi 2770	. . . . . . . . 9 ⊢ 2 = (2↑1)
39		4cn 12325	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
40		ax-1cn 11157	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
41		4p1e5 12385	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
42	39, 40, 41	mvlladdi 11475	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
43	42	oveq2i 7421	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
44	38, 43	eqtri 2784	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
45		4z 12627	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
46	18, 45	pm3.2i 475	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
47		expsub 14145	. . . . . . . . 9 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (5 ∈ ℤ ∧ 4 ∈ ℤ)) → (2↑(5 − 4)) = ((2↑5) / (2↑4)))
48	6, 46, 47	mp2an 704	. . . . . . . 8 ⊢ (2↑(5 − 4)) = ((2↑5) / (2↑4))
49		2exp4 17143	. . . . . . . . 9 ⊢ (2↑4) = ;16
50	49	oveq2i 7421	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
51	44, 48, 50	3eqtri 2788	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
52	51	eqcomi 2770	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
53	52	oveq2i 7421	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
54	10, 35, 53	3eqtri 2788	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
55		3nn0 12521	. . . . . . 7 ⊢ 3 ∈ ℕ₀
56		expdiv 14148	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
57	5, 6, 55, 56	mp3an 1488	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
58	57	oveq2i 7421	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
59		5t4e20 12817	. . . . . . . 8 ⊢ (5 · 4) = ;20
60	59	eqcomi 2770	. . . . . . 7 ⊢ ;20 = (5 · 4)
61	60	oveq1i 7420	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
62		5cn 12328	. . . . . . 7 ⊢ 5 ∈ ℂ
63		expcl 14114	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
64	5, 55, 63	mp2an 704	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
65		expcl 14114	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
66	13, 55, 65	mp2an 704	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
67		3z 12626	. . . . . . . . 9 ⊢ 3 ∈ ℤ
68		expne0i 14129	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
69	13, 16, 67, 68	mp3an 1488	. . . . . . . 8 ⊢ (2↑3) ≠ 0
70	64, 66, 69	divcli 11956	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
71	62, 39, 70	mulassi 11219	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
72	61, 71	eqtri 2784	. . . . 5 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
73	66, 69	pm3.2i 475	. . . . . . . . 9 ⊢ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0)
74		4ne0 12351	. . . . . . . . . 10 ⊢ 4 ≠ 0
75	39, 74	pm3.2i 475	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
76		divdiv2 11926	. . . . . . . . 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 1488	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
78	64, 39	mulcomi 11216	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
79	78	oveq1i 7420	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
80	39, 64, 66, 69	divassi 11970	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
81	77, 79, 80	3eqtrri 2789	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
82		4t2e8 12408	. . . . . . . . . 10 ⊢ (4 · 2) = 8
83		cu2 14235	. . . . . . . . . . 11 ⊢ (2↑3) = 8
84	83	eqcomi 2770	. . . . . . . . . 10 ⊢ 8 = (2↑3)
85	82, 84	eqtri 2784	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
86	66, 39, 13, 74	divmuli 11968	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
87	85, 86	mpbir 234	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
88	87	oveq2i 7421	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
89	81, 88	eqtri 2784	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
90	89	oveq2i 7421	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
91	58, 72, 90	3eqtri 2788	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
92	54, 91	oveq12i 7422	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
93	92	oveq1i 7420	. 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 2792	1 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 − cmin 11440 -cneg 11441 / cdiv 11870 2c2 12294 3c3 12295 4c4 12296 5c5 12297 6c6 12298 8c8 12300 ℕ₀cn0 12503 ℤcz 12590 cdc 12710 ↑cexp 14096 cosccos 16117 πcpi 16119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-seq 14037 df-exp 14097 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15103 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-limsup 15521 df-clim 15538 df-rlim 15539 df-sum 15737 df-ef 16120 df-sin 16122 df-cos 16123 df-pi 16125 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-lp 23272 df-perf 23273 df-cn 23363 df-cnp 23364 df-haus 23451 df-tx 23698 df-hmeo 23891 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-tms 24458 df-cncf 25016 df-limc 26004 df-dv 26005

This theorem is referenced by: (None)

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