goldratmolem2 - Mathbox for Ender Ting

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Theorem goldratmolem2 48136

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 48135	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26828	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 48131	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11347	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12579	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12650	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14280	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1491	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7451	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14246	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 705	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12442	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14246	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 705	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12473	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12453	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12744	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14261	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1491	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 476	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		16nn0 12856	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
23	22	nn0cni 12642	. . . . . . . 8 ⊢ ;16 ∈ ℂ
24		1nn0 12646	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
25		6nn 12456	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
26	24, 25	decnncl 12862	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
27	26	nnne0i 12402	. . . . . . . 8 ⊢ ;16 ≠ 0
28	23, 27	pm3.2i 476	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
29		divdiv2 12053	. . . . . . 7 ⊢ (((𝐹↑5) ∈ ℂ ∧ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0) ∧ (;16 ∈ ℂ ∧ ;16 ≠ 0)) → ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5)))
30	12, 21, 28, 29	mp3an 1491	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
31	12, 23	mulcomi 11341	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
32	31	oveq1i 7450	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
33	23, 12, 15, 20	divassi 12097	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
34	30, 32, 33	3eqtrri 2791	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
35		exp1 14234	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
36	13, 35	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
37	36	eqcomi 2772	. . . . . . . . 9 ⊢ 2 = (2↑1)
38		4cn 12452	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
39		ax-1cn 11282	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
40		4p1e5 12512	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
41	38, 39, 40	mvlladdi 11600	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
42	41	oveq2i 7451	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
43	37, 42	eqtri 2786	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
44		4z 12754	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
45	18, 44	pm3.2i 476	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
46		expsub 14277	. . . . . . . . 9 ⊢ (((2 ∈ ℂ ∧ 2 ≠ 0) ∧ (5 ∈ ℤ ∧ 4 ∈ ℤ)) → (2↑(5 − 4)) = ((2↑5) / (2↑4)))
47	6, 45, 46	mp2an 705	. . . . . . . 8 ⊢ (2↑(5 − 4)) = ((2↑5) / (2↑4))
48		2exp4 17288	. . . . . . . . 9 ⊢ (2↑4) = ;16
49	48	oveq2i 7451	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
50	43, 47, 49	3eqtri 2790	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
51	50	eqcomi 2772	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
52	51	oveq2i 7451	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
53	10, 34, 52	3eqtri 2790	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
54		3nn0 12648	. . . . . . 7 ⊢ 3 ∈ ℕ₀
55		expdiv 14280	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
56	5, 6, 54, 55	mp3an 1491	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
57	56	oveq2i 7451	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
58		5t4e20 12945	. . . . . . . 8 ⊢ (5 · 4) = ;20
59	58	eqcomi 2772	. . . . . . 7 ⊢ ;20 = (5 · 4)
60	59	oveq1i 7450	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
61		5cn 12455	. . . . . . 7 ⊢ 5 ∈ ℂ
62		expcl 14246	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
63	5, 54, 62	mp2an 705	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
64		expcl 14246	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
65	13, 54, 64	mp2an 705	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
66		3z 12753	. . . . . . . . 9 ⊢ 3 ∈ ℤ
67		expne0i 14261	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
68	13, 16, 66, 67	mp3an 1491	. . . . . . . 8 ⊢ (2↑3) ≠ 0
69	63, 65, 68	divcli 12083	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
70	61, 38, 69	mulassi 11344	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
71	60, 70	eqtri 2786	. . . . 5 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
72	65, 68	pm3.2i 476	. . . . . . . . 9 ⊢ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0)
73		4ne0 12478	. . . . . . . . . 10 ⊢ 4 ≠ 0
74	38, 73	pm3.2i 476	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
75		divdiv2 12053	. . . . . . . . 9 ⊢ (((𝐹↑3) ∈ ℂ ∧ ((2↑3) ∈ ℂ ∧ (2↑3) ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3)))
76	63, 72, 74, 75	mp3an 1491	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
77	63, 38	mulcomi 11341	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
78	77	oveq1i 7450	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
79	38, 63, 65, 68	divassi 12097	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
80	76, 78, 79	3eqtrri 2791	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
81		4t2e8 12535	. . . . . . . . . 10 ⊢ (4 · 2) = 8
82		cu2 14367	. . . . . . . . . . 11 ⊢ (2↑3) = 8
83	82	eqcomi 2772	. . . . . . . . . 10 ⊢ 8 = (2↑3)
84	81, 83	eqtri 2786	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
85	65, 38, 13, 73	divmuli 12095	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
86	84, 85	mpbir 234	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
87	86	oveq2i 7451	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
88	80, 87	eqtri 2786	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
89	88	oveq2i 7451	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
90	57, 71, 89	3eqtri 2790	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
91	53, 90	oveq12i 7452	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
92	91	oveq1i 7450	. 2 ⊢ (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2))) = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))
93	2, 3, 92	3eqtr3i 2794	1 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ‘cfv 6555 (class class class)co 7440 ℂcc 11222 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 − cmin 11565 -cneg 11566 / cdiv 11997 2c2 12421 3c3 12422 4c4 12423 5c5 12424 6c6 12425 8c8 12427 ℕ₀cn0 12630 ℤcz 12717 cdc 12838 ↑cexp 14228 cosccos 16256 πcpi 16258

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-addf 11303

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-fi 9427 df-sup 9458 df-inf 9459 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-ioc 13505 df-ico 13506 df-icc 13507 df-fz 13664 df-fzo 13813 df-fl 13956 df-seq 14169 df-exp 14229 df-fac 14442 df-bc 14471 df-hash 14499 df-shft 15246 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-limsup 15664 df-clim 15681 df-rlim 15682 df-sum 15880 df-ef 16259 df-sin 16261 df-cos 16262 df-pi 16264 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-pt 17641 df-prds 17644 df-xrs 17700 df-qtop 17705 df-imas 17706 df-xps 17708 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-mulg 19304 df-cntz 19558 df-cmn 20023 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-fbas 21702 df-fg 21703 df-cnfld 21706 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-cld 23364 df-ntr 23365 df-cls 23366 df-nei 23443 df-lp 23481 df-perf 23482 df-cn 23572 df-cnp 23573 df-haus 23660 df-tx 23908 df-hmeo 24101 df-fil 24192 df-fm 24284 df-flim 24285 df-flf 24286 df-xms 24666 df-ms 24667 df-tms 24668 df-cncf 25226 df-limc 26213 df-dv 26214

This theorem is used by: goldratmolem3 48137

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