goldratval - Mathbox for Ender Ting

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Theorem goldratval 47819

Description: Value of the golden ratio. (Contributed by Ender Ting, 24-Jul-2026.)

**Hypothesis**
Ref	Expression
goldra.val	⊢ 𝐹 = (2 · (cos‘(π / 5)))

**Assertion**
Ref	Expression
goldratval	⊢ 𝐹 = ((1 + (√‘5)) / 2)

**Proof of Theorem goldratval**
Step	Hyp	Ref	Expression
1		1lt5 12472	. . . . . . . . . 10 ⊢ 1 < 5
2		0le1 11786	. . . . . . . . . . 11 ⊢ 0 ≤ 1
3		5nn0 12573	. . . . . . . . . . . 12 ⊢ 5 ∈ ℕ₀
4	3	nn0ge0i 12580	. . . . . . . . . . 11 ⊢ 0 ≤ 5
5		1re 11257	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
6		5re 12377	. . . . . . . . . . . 12 ⊢ 5 ∈ ℝ
7	5, 6	sqrtlti 15502	. . . . . . . . . . 11 ⊢ ((0 ≤ 1 ∧ 0 ≤ 5) → (1 < 5 ↔ (√‘1) < (√‘5)))
8	2, 4, 7	mp2an 705	. . . . . . . . . 10 ⊢ (1 < 5 ↔ (√‘1) < (√‘5))
9	1, 8	mpbi 233	. . . . . . . . 9 ⊢ (√‘1) < (√‘5)
10		negneg1e1 12256	. . . . . . . . . 10 ⊢ --1 = 1
11		sqrt1 15383	. . . . . . . . . 10 ⊢ (√‘1) = 1
12	10, 11	eqtr4i 2786	. . . . . . . . 9 ⊢ --1 = (√‘1)
13		5pos 12402	. . . . . . . . . . . 12 ⊢ 0 < 5
14	6, 13	sqrtpclii 15495	. . . . . . . . . . 11 ⊢ (√‘5) ∈ ℝ
15	14	recni 11272	. . . . . . . . . 10 ⊢ (√‘5) ∈ ℂ
16	15	addridi 11446	. . . . . . . . 9 ⊢ ((√‘5) + 0) = (√‘5)
17	9, 12, 16	3brtr4i 5135	. . . . . . . 8 ⊢ --1 < ((√‘5) + 0)
18		neg1rr 12253	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
19	18	renegcli 11568	. . . . . . . . 9 ⊢ --1 ∈ ℝ
20		0re 11259	. . . . . . . . 9 ⊢ 0 ∈ ℝ
21	19, 14, 20	ltsubadd2i 11822	. . . . . . . 8 ⊢ ((--1 − (√‘5)) < 0 ↔ --1 < ((√‘5) + 0))
22	17, 21	mpbir 234	. . . . . . 7 ⊢ (--1 − (√‘5)) < 0
23	19, 14	resubcli 11569	. . . . . . . 8 ⊢ (--1 − (√‘5)) ∈ ℝ
24		2re 12364	. . . . . . . . 9 ⊢ 2 ∈ ℝ
25	24, 5	remulcli 11274	. . . . . . . 8 ⊢ (2 · 1) ∈ ℝ
26		2pos 12394	. . . . . . . . 9 ⊢ 0 < 2
27		2t1e2 12452	. . . . . . . . 9 ⊢ (2 · 1) = 2
28	26, 27	breqtrri 5132	. . . . . . . 8 ⊢ 0 < (2 · 1)
29	23, 20, 25, 28	ltdiv1ii 12193	. . . . . . 7 ⊢ ((--1 − (√‘5)) < 0 ↔ ((--1 − (√‘5)) / (2 · 1)) < (0 / (2 · 1)))
30	22, 29	mpbi 233	. . . . . 6 ⊢ ((--1 − (√‘5)) / (2 · 1)) < (0 / (2 · 1))
31	25	recni 11272	. . . . . . 7 ⊢ (2 · 1) ∈ ℂ
32	20, 28	gtneii 11371	. . . . . . 7 ⊢ (2 · 1) ≠ 0
33	31, 32	div0i 11998	. . . . . 6 ⊢ (0 / (2 · 1)) = 0
34	30, 33	breqtri 5130	. . . . 5 ⊢ ((--1 − (√‘5)) / (2 · 1)) < 0
35	23, 25, 32	redivcli 12031	. . . . . 6 ⊢ ((--1 − (√‘5)) / (2 · 1)) ∈ ℝ
36	35, 20	ltnsymi 11378	. . . . 5 ⊢ (((--1 − (√‘5)) / (2 · 1)) < 0 → ¬ 0 < ((--1 − (√‘5)) / (2 · 1)))
37	34, 36	ax-mp 5	. . . 4 ⊢ ¬ 0 < ((--1 − (√‘5)) / (2 · 1))
38		goldra.val	. . . . . 6 ⊢ 𝐹 = (2 · (cos‘(π / 5)))
39	38	goldrapos 47813	. . . . 5 ⊢ 0 < 𝐹
40		breq2 5107	. . . . 5 ⊢ (𝐹 = ((--1 − (√‘5)) / (2 · 1)) → (0 < 𝐹 ↔ 0 < ((--1 − (√‘5)) / (2 · 1))))
41	39, 40	mpbii 236	. . . 4 ⊢ (𝐹 = ((--1 − (√‘5)) / (2 · 1)) → 0 < ((--1 − (√‘5)) / (2 · 1)))
42	37, 41	mto 200	. . 3 ⊢ ¬ 𝐹 = ((--1 − (√‘5)) / (2 · 1))
43	38	goldrarr 47811	. . . . . . . . . 10 ⊢ 𝐹 ∈ ℝ
44	43	recni 11272	. . . . . . . . 9 ⊢ 𝐹 ∈ ℂ
45	44	sqcli 14270	. . . . . . . 8 ⊢ (𝐹↑2) ∈ ℂ
46		ax-1cn 11207	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47	44, 46	addcli 11264	. . . . . . . 8 ⊢ (𝐹 + 1) ∈ ℂ
48	45, 47	negsubi 11585	. . . . . . 7 ⊢ ((𝐹↑2) + -(𝐹 + 1)) = ((𝐹↑2) − (𝐹 + 1))
49	45	mullidi 11263	. . . . . . . 8 ⊢ (1 · (𝐹↑2)) = (𝐹↑2)
50	44	mulm1i 11708	. . . . . . . . . 10 ⊢ (-1 · 𝐹) = -𝐹
51	50	oveq1i 7426	. . . . . . . . 9 ⊢ ((-1 · 𝐹) + -1) = (-𝐹 + -1)
52	44, 46	negdii 11591	. . . . . . . . 9 ⊢ -(𝐹 + 1) = (-𝐹 + -1)
53	51, 52	eqtr4i 2786	. . . . . . . 8 ⊢ ((-1 · 𝐹) + -1) = -(𝐹 + 1)
54	49, 53	oveq12i 7428	. . . . . . 7 ⊢ ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = ((𝐹↑2) + -(𝐹 + 1))
55		subsub4 11540	. . . . . . . 8 ⊢ (((𝐹↑2) ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐹↑2) − 𝐹) − 1) = ((𝐹↑2) − (𝐹 + 1)))
56	45, 44, 46, 55	mp3an 1490	. . . . . . 7 ⊢ (((𝐹↑2) − 𝐹) − 1) = ((𝐹↑2) − (𝐹 + 1))
57	48, 54, 56	3eqtr4ri 2794	. . . . . 6 ⊢ (((𝐹↑2) − 𝐹) − 1) = ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1))
58	38	goldratmolem4 47818	. . . . . 6 ⊢ (((𝐹↑2) − 𝐹) − 1) = 0
59	57, 58	eqtr3i 2785	. . . . 5 ⊢ ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = 0
60		1cnd 11251	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
61		ax-1ne0 11218	. . . . . . . 8 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ≠ 0)
63		neg1cn 12252	. . . . . . . 8 ⊢ -1 ∈ ℂ
64	63	a1i 11	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℂ)
65	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 ∈ ℂ)
66		4cn 12375	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
67	46, 66	subnegi 11586	. . . . . . . . 9 ⊢ (1 − -4) = (1 + 4)
68		neg1sqe1 14285	. . . . . . . . . 10 ⊢ (-1↑2) = 1
69	63	mullidi 11263	. . . . . . . . . . . 12 ⊢ (1 · -1) = -1
70	69	oveq2i 7427	. . . . . . . . . . 11 ⊢ (4 · (1 · -1)) = (4 · -1)
71	66, 46	mulneg2i 11710	. . . . . . . . . . 11 ⊢ (4 · -1) = -(4 · 1)
72	66	mulridi 11262	. . . . . . . . . . . 12 ⊢ (4 · 1) = 4
73	72	negeqi 11499	. . . . . . . . . . 11 ⊢ -(4 · 1) = -4
74	70, 71, 73	3eqtri 2787	. . . . . . . . . 10 ⊢ (4 · (1 · -1)) = -4
75	68, 74	oveq12i 7428	. . . . . . . . 9 ⊢ ((-1↑2) − (4 · (1 · -1))) = (1 − -4)
76		df-5 12355	. . . . . . . . . 10 ⊢ 5 = (4 + 1)
77	66, 46, 76	comraddi 11474	. . . . . . . . 9 ⊢ 5 = (1 + 4)
78	67, 75, 77	3eqtr4ri 2794	. . . . . . . 8 ⊢ 5 = ((-1↑2) − (4 · (1 · -1)))
79	78	a1i 11	. . . . . . 7 ⊢ (⊤ → 5 = ((-1↑2) − (4 · (1 · -1))))
80	60, 62, 64, 64, 65, 79	quad 27109	. . . . . 6 ⊢ (⊤ → (((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = 0 ↔ (𝐹 = ((--1 + (√‘5)) / (2 · 1)) ∨ 𝐹 = ((--1 − (√‘5)) / (2 · 1)))))
81	80	mptru 1577	. . . . 5 ⊢ (((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = 0 ↔ (𝐹 = ((--1 + (√‘5)) / (2 · 1)) ∨ 𝐹 = ((--1 − (√‘5)) / (2 · 1))))
82	59, 81	mpbi 233	. . . 4 ⊢ (𝐹 = ((--1 + (√‘5)) / (2 · 1)) ∨ 𝐹 = ((--1 − (√‘5)) / (2 · 1)))
83	82	ori 875	. . 3 ⊢ (¬ 𝐹 = ((--1 + (√‘5)) / (2 · 1)) → 𝐹 = ((--1 − (√‘5)) / (2 · 1)))
84	42, 83	mt3 204	. 2 ⊢ 𝐹 = ((--1 + (√‘5)) / (2 · 1))
85	10	oveq1i 7426	. . 3 ⊢ (--1 + (√‘5)) = (1 + (√‘5))
86	85, 27	oveq12i 7428	. 2 ⊢ ((--1 + (√‘5)) / (2 · 1)) = ((1 + (√‘5)) / 2)
87	84, 86	eqtri 2783	1 ⊢ 𝐹 = ((1 + (√‘5)) / 2)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∨ wo 861 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 < clt 11292 ≤ cle 11293 − cmin 11490 -cneg 11491 / cdiv 11920 2c2 12344 4c4 12346 5c5 12347 ↑cexp 14150 √csqrt 15345 cosccos 16175 πcpi 16177

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138 df-limc 26125 df-dv 26126

This theorem is used by: (None)

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