goldratval - Mathbox for Ender Ting

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Theorem goldratval 47739

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 12448	. . . . . . . . . 10 ⊢ 1 < 5
2		0le1 11762	. . . . . . . . . . 11 ⊢ 0 ≤ 1
3		5nn0 12549	. . . . . . . . . . . 12 ⊢ 5 ∈ ℕ₀
4	3	nn0ge0i 12556	. . . . . . . . . . 11 ⊢ 0 ≤ 5
5		1re 11233	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
6		5re 12353	. . . . . . . . . . . 12 ⊢ 5 ∈ ℝ
7	5, 6	sqrtlti 15477	. . . . . . . . . . 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 12232	. . . . . . . . . 10 ⊢ --1 = 1
11		sqrt1 15358	. . . . . . . . . 10 ⊢ (√‘1) = 1
12	10, 11	eqtr4i 2788	. . . . . . . . 9 ⊢ --1 = (√‘1)
13		5pos 12378	. . . . . . . . . . . 12 ⊢ 0 < 5
14	6, 13	sqrtpclii 15470	. . . . . . . . . . 11 ⊢ (√‘5) ∈ ℝ
15	14	recni 11248	. . . . . . . . . 10 ⊢ (√‘5) ∈ ℂ
16	15	addridi 11422	. . . . . . . . 9 ⊢ ((√‘5) + 0) = (√‘5)
17	9, 12, 16	3brtr4i 5139	. . . . . . . 8 ⊢ --1 < ((√‘5) + 0)
18		neg1rr 12229	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
19	18	renegcli 11544	. . . . . . . . 9 ⊢ --1 ∈ ℝ
20		0re 11235	. . . . . . . . 9 ⊢ 0 ∈ ℝ
21	19, 14, 20	ltsubadd2i 11798	. . . . . . . 8 ⊢ ((--1 − (√‘5)) < 0 ↔ --1 < ((√‘5) + 0))
22	17, 21	mpbir 234	. . . . . . 7 ⊢ (--1 − (√‘5)) < 0
23	19, 14	resubcli 11545	. . . . . . . 8 ⊢ (--1 − (√‘5)) ∈ ℝ
24		2re 12340	. . . . . . . . 9 ⊢ 2 ∈ ℝ
25	24, 5	remulcli 11250	. . . . . . . 8 ⊢ (2 · 1) ∈ ℝ
26		2pos 12370	. . . . . . . . 9 ⊢ 0 < 2
27		2t1e2 12428	. . . . . . . . 9 ⊢ (2 · 1) = 2
28	26, 27	breqtrri 5136	. . . . . . . 8 ⊢ 0 < (2 · 1)
29	23, 20, 25, 28	ltdiv1ii 12169	. . . . . . 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 11248	. . . . . . 7 ⊢ (2 · 1) ∈ ℂ
32	20, 28	gtneii 11347	. . . . . . 7 ⊢ (2 · 1) ≠ 0
33	31, 32	div0i 11974	. . . . . 6 ⊢ (0 / (2 · 1)) = 0
34	30, 33	breqtri 5134	. . . . 5 ⊢ ((--1 − (√‘5)) / (2 · 1)) < 0
35	23, 25, 32	redivcli 12007	. . . . . 6 ⊢ ((--1 − (√‘5)) / (2 · 1)) ∈ ℝ
36	35, 20	ltnsymi 11354	. . . . 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 47733	. . . . 5 ⊢ 0 < 𝐹
40		breq2 5111	. . . . 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 47731	. . . . . . . . . 10 ⊢ 𝐹 ∈ ℝ
44	43	recni 11248	. . . . . . . . 9 ⊢ 𝐹 ∈ ℂ
45	44	sqcli 14245	. . . . . . . 8 ⊢ (𝐹↑2) ∈ ℂ
46		ax-1cn 11183	. . . . . . . . 9 ⊢ 1 ∈ ℂ
47	44, 46	addcli 11240	. . . . . . . 8 ⊢ (𝐹 + 1) ∈ ℂ
48	45, 47	negsubi 11561	. . . . . . 7 ⊢ ((𝐹↑2) + -(𝐹 + 1)) = ((𝐹↑2) − (𝐹 + 1))
49	45	mullidi 11239	. . . . . . . 8 ⊢ (1 · (𝐹↑2)) = (𝐹↑2)
50	44	mulm1i 11684	. . . . . . . . . 10 ⊢ (-1 · 𝐹) = -𝐹
51	50	oveq1i 7426	. . . . . . . . 9 ⊢ ((-1 · 𝐹) + -1) = (-𝐹 + -1)
52	44, 46	negdii 11567	. . . . . . . . 9 ⊢ -(𝐹 + 1) = (-𝐹 + -1)
53	51, 52	eqtr4i 2788	. . . . . . . 8 ⊢ ((-1 · 𝐹) + -1) = -(𝐹 + 1)
54	49, 53	oveq12i 7428	. . . . . . 7 ⊢ ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = ((𝐹↑2) + -(𝐹 + 1))
55		subsub4 11516	. . . . . . . 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 2796	. . . . . 6 ⊢ (((𝐹↑2) − 𝐹) − 1) = ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1))
58	38	goldratmolem4 47738	. . . . . 6 ⊢ (((𝐹↑2) − 𝐹) − 1) = 0
59	57, 58	eqtr3i 2787	. . . . 5 ⊢ ((1 · (𝐹↑2)) + ((-1 · 𝐹) + -1)) = 0
60		1cnd 11227	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
61		ax-1ne0 11194	. . . . . . . 8 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ≠ 0)
63		neg1cn 12228	. . . . . . . 8 ⊢ -1 ∈ ℂ
64	63	a1i 11	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℂ)
65	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 ∈ ℂ)
66		4cn 12351	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
67	46, 66	subnegi 11562	. . . . . . . . 9 ⊢ (1 − -4) = (1 + 4)
68		neg1sqe1 14260	. . . . . . . . . 10 ⊢ (-1↑2) = 1
69	63	mullidi 11239	. . . . . . . . . . . 12 ⊢ (1 · -1) = -1
70	69	oveq2i 7427	. . . . . . . . . . 11 ⊢ (4 · (1 · -1)) = (4 · -1)
71	66, 46	mulneg2i 11686	. . . . . . . . . . 11 ⊢ (4 · -1) = -(4 · 1)
72	66	mulridi 11238	. . . . . . . . . . . 12 ⊢ (4 · 1) = 4
73	72	negeqi 11475	. . . . . . . . . . 11 ⊢ -(4 · 1) = -4
74	70, 71, 73	3eqtri 2789	. . . . . . . . . 10 ⊢ (4 · (1 · -1)) = -4
75	68, 74	oveq12i 7428	. . . . . . . . 9 ⊢ ((-1↑2) − (4 · (1 · -1))) = (1 − -4)
76		df-5 12331	. . . . . . . . . 10 ⊢ 5 = (4 + 1)
77	66, 46, 76	comraddi 11450	. . . . . . . . 9 ⊢ 5 = (1 + 4)
78	67, 75, 77	3eqtr4ri 2796	. . . . . . . 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 27073	. . . . . 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 2785	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 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 ℂcc 11123 0cc0 11125 1c1 11126 + caddc 11128 · cmul 11130 < clt 11268 ≤ cle 11269 − cmin 11466 -cneg 11467 / cdiv 11896 2c2 12320 4c4 12322 5c5 12323 ↑cexp 14125 √csqrt 15320 cosccos 16152 πcpi 16154

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 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

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 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16155 df-sin 16157 df-cos 16158 df-pi 16160 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-rest 17509 df-topn 17510 df-0g 17528 df-gsum 17529 df-topgen 17530 df-pt 17531 df-prds 17534 df-xrs 17590 df-qtop 17595 df-imas 17596 df-xps 17598 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-submnd 18891 df-mulg 19190 df-cntz 19443 df-cmn 19908 df-psmet 21576 df-xmet 21577 df-met 21578 df-bl 21579 df-mopn 21580 df-fbas 21581 df-fg 21582 df-cnfld 21585 df-top 23118 df-topon 23135 df-topsp 23157 df-bases 23170 df-cld 23243 df-ntr 23244 df-cls 23245 df-nei 23322 df-lp 23360 df-perf 23361 df-cn 23451 df-cnp 23452 df-haus 23539 df-tx 23787 df-hmeo 23980 df-fil 24071 df-fm 24163 df-flim 24164 df-flf 24165 df-xms 24545 df-ms 24546 df-tms 24547 df-cncf 25105 df-limc 26093 df-dv 26094

This theorem is used by: (None)

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