goldratmolem2 - Mathbox for Ender Ting

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Theorem goldratmolem2 47738

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 47737	. 2 ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
3		cospi 26707	. 2 ⊢ (cos‘π) = -1
4	1	goldrarr 47733	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
5	4	recni 11250	. . . . . . 7 ⊢ 𝐹 ∈ ℂ
6		2cnne0 12480	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
7		5nn0 12551	. . . . . . 7 ⊢ 5 ∈ ℕ₀
8		expdiv 14179	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 5 ∈ ℕ₀) → ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5)))
9	5, 6, 7, 8	mp3an 1490	. . . . . 6 ⊢ ((𝐹 / 2)↑5) = ((𝐹↑5) / (2↑5))
10	9	oveq2i 7427	. . . . 5 ⊢ (;16 · ((𝐹 / 2)↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
11		expcl 14145	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
12	5, 7, 11	mp2an 705	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
13		2cn 12343	. . . . . . . . 9 ⊢ 2 ∈ ℂ
14		expcl 14145	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 5 ∈ ℕ₀) → (2↑5) ∈ ℂ)
15	13, 7, 14	mp2an 705	. . . . . . . 8 ⊢ (2↑5) ∈ ℂ
16		2ne0 12374	. . . . . . . . 9 ⊢ 2 ≠ 0
17		5nn 12354	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
18	17	nnzi 12645	. . . . . . . . 9 ⊢ 5 ∈ ℤ
19		expne0i 14160	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 5 ∈ ℤ) → (2↑5) ≠ 0)
20	13, 16, 18, 19	mp3an 1490	. . . . . . . 8 ⊢ (2↑5) ≠ 0
21	15, 20	pm3.2i 476	. . . . . . 7 ⊢ ((2↑5) ∈ ℂ ∧ (2↑5) ≠ 0)
22		16nn0 12757	. . . . . . . . 9 ⊢ ;16 ∈ ℕ₀
23	22	nn0cni 12543	. . . . . . . 8 ⊢ ;16 ∈ ℂ
24		1nn0 12547	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
25		6nn 12357	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
26	24, 25	decnncl 12763	. . . . . . . . 9 ⊢ ;16 ∈ ℕ
27	26	nnne0i 12303	. . . . . . . 8 ⊢ ;16 ≠ 0
28	23, 27	pm3.2i 476	. . . . . . 7 ⊢ (;16 ∈ ℂ ∧ ;16 ≠ 0)
29		divdiv2 11954	. . . . . . 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 1490	. . . . . 6 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = (((𝐹↑5) · ;16) / (2↑5))
31	12, 23	mulcomi 11244	. . . . . . 7 ⊢ ((𝐹↑5) · ;16) = (;16 · (𝐹↑5))
32	31	oveq1i 7426	. . . . . 6 ⊢ (((𝐹↑5) · ;16) / (2↑5)) = ((;16 · (𝐹↑5)) / (2↑5))
33	23, 12, 15, 20	divassi 11998	. . . . . 6 ⊢ ((;16 · (𝐹↑5)) / (2↑5)) = (;16 · ((𝐹↑5) / (2↑5)))
34	30, 32, 33	3eqtrri 2790	. . . . 5 ⊢ (;16 · ((𝐹↑5) / (2↑5))) = ((𝐹↑5) / ((2↑5) / ;16))
35		exp1 14133	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
36	13, 35	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
37	36	eqcomi 2771	. . . . . . . . 9 ⊢ 2 = (2↑1)
38		4cn 12353	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
39		ax-1cn 11185	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
40		4p1e5 12413	. . . . . . . . . . 11 ⊢ (4 + 1) = 5
41	38, 39, 40	mvlladdi 11503	. . . . . . . . . 10 ⊢ 1 = (5 − 4)
42	41	oveq2i 7427	. . . . . . . . 9 ⊢ (2↑1) = (2↑(5 − 4))
43	37, 42	eqtri 2785	. . . . . . . 8 ⊢ 2 = (2↑(5 − 4))
44		4z 12655	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
45	18, 44	pm3.2i 476	. . . . . . . . 9 ⊢ (5 ∈ ℤ ∧ 4 ∈ ℤ)
46		expsub 14176	. . . . . . . . 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 17180	. . . . . . . . 9 ⊢ (2↑4) = ;16
49	48	oveq2i 7427	. . . . . . . 8 ⊢ ((2↑5) / (2↑4)) = ((2↑5) / ;16)
50	43, 47, 49	3eqtri 2789	. . . . . . 7 ⊢ 2 = ((2↑5) / ;16)
51	50	eqcomi 2771	. . . . . 6 ⊢ ((2↑5) / ;16) = 2
52	51	oveq2i 7427	. . . . 5 ⊢ ((𝐹↑5) / ((2↑5) / ;16)) = ((𝐹↑5) / 2)
53	10, 34, 52	3eqtri 2789	. . . 4 ⊢ (;16 · ((𝐹 / 2)↑5)) = ((𝐹↑5) / 2)
54		3nn0 12549	. . . . . . 7 ⊢ 3 ∈ ℕ₀
55		expdiv 14179	. . . . . . 7 ⊢ ((𝐹 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ 3 ∈ ℕ₀) → ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3)))
56	5, 6, 54, 55	mp3an 1490	. . . . . 6 ⊢ ((𝐹 / 2)↑3) = ((𝐹↑3) / (2↑3))
57	56	oveq2i 7427	. . . . 5 ⊢ (;20 · ((𝐹 / 2)↑3)) = (;20 · ((𝐹↑3) / (2↑3)))
58		5t4e20 12846	. . . . . . . 8 ⊢ (5 · 4) = ;20
59	58	eqcomi 2771	. . . . . . 7 ⊢ ;20 = (5 · 4)
60	59	oveq1i 7426	. . . . . 6 ⊢ (;20 · ((𝐹↑3) / (2↑3))) = ((5 · 4) · ((𝐹↑3) / (2↑3)))
61		5cn 12356	. . . . . . 7 ⊢ 5 ∈ ℂ
62		expcl 14145	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
63	5, 54, 62	mp2an 705	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
64		expcl 14145	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ₀) → (2↑3) ∈ ℂ)
65	13, 54, 64	mp2an 705	. . . . . . . 8 ⊢ (2↑3) ∈ ℂ
66		3z 12654	. . . . . . . . 9 ⊢ 3 ∈ ℤ
67		expne0i 14160	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 3 ∈ ℤ) → (2↑3) ≠ 0)
68	13, 16, 66, 67	mp3an 1490	. . . . . . . 8 ⊢ (2↑3) ≠ 0
69	63, 65, 68	divcli 11984	. . . . . . 7 ⊢ ((𝐹↑3) / (2↑3)) ∈ ℂ
70	61, 38, 69	mulassi 11247	. . . . . 6 ⊢ ((5 · 4) · ((𝐹↑3) / (2↑3))) = (5 · (4 · ((𝐹↑3) / (2↑3))))
71	60, 70	eqtri 2785	. . . . 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 12379	. . . . . . . . . 10 ⊢ 4 ≠ 0
74	38, 73	pm3.2i 476	. . . . . . . . 9 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
75		divdiv2 11954	. . . . . . . . 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 1490	. . . . . . . 8 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = (((𝐹↑3) · 4) / (2↑3))
77	63, 38	mulcomi 11244	. . . . . . . . 9 ⊢ ((𝐹↑3) · 4) = (4 · (𝐹↑3))
78	77	oveq1i 7426	. . . . . . . 8 ⊢ (((𝐹↑3) · 4) / (2↑3)) = ((4 · (𝐹↑3)) / (2↑3))
79	38, 63, 65, 68	divassi 11998	. . . . . . . 8 ⊢ ((4 · (𝐹↑3)) / (2↑3)) = (4 · ((𝐹↑3) / (2↑3)))
80	76, 78, 79	3eqtrri 2790	. . . . . . 7 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / ((2↑3) / 4))
81		4t2e8 12436	. . . . . . . . . 10 ⊢ (4 · 2) = 8
82		cu2 14266	. . . . . . . . . . 11 ⊢ (2↑3) = 8
83	82	eqcomi 2771	. . . . . . . . . 10 ⊢ 8 = (2↑3)
84	81, 83	eqtri 2785	. . . . . . . . 9 ⊢ (4 · 2) = (2↑3)
85	65, 38, 13, 73	divmuli 11996	. . . . . . . . 9 ⊢ (((2↑3) / 4) = 2 ↔ (4 · 2) = (2↑3))
86	84, 85	mpbir 234	. . . . . . . 8 ⊢ ((2↑3) / 4) = 2
87	86	oveq2i 7427	. . . . . . 7 ⊢ ((𝐹↑3) / ((2↑3) / 4)) = ((𝐹↑3) / 2)
88	80, 87	eqtri 2785	. . . . . 6 ⊢ (4 · ((𝐹↑3) / (2↑3))) = ((𝐹↑3) / 2)
89	88	oveq2i 7427	. . . . 5 ⊢ (5 · (4 · ((𝐹↑3) / (2↑3)))) = (5 · ((𝐹↑3) / 2))
90	57, 71, 89	3eqtri 2789	. . . 4 ⊢ (;20 · ((𝐹 / 2)↑3)) = (5 · ((𝐹↑3) / 2))
91	53, 90	oveq12i 7428	. . 3 ⊢ ((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) = (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2)))
92	91	oveq1i 7426	. 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 2793	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 2145 ≠ wne 2957 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 -cneg 11469 / cdiv 11898 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 8c8 12328 ℕ₀cn0 12531 ℤcz 12618 cdc 12739 ↑cexp 14127 cosccos 16154 πcpi 16156

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-shft 15142 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776 df-ef 16157 df-sin 16159 df-cos 16160 df-pi 16162 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-limc 26095 df-dv 26096

This theorem is used by: goldratmolem3 47739

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