goldratmolem3 - Mathbox for Ender Ting

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Theorem goldratmolem3 48137

Description: Lemma 3 for determining the value of golden ratio. (Contributed by Ender Ting, 23-Jul-2026.)

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

**Assertion**
Ref	Expression
goldratmolem3	⊢ ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0

**Proof of Theorem goldratmolem3**
Step	Hyp	Ref	Expression
1		goldra.val	. . . . 5 ⊢ 𝐹 = (2 · (cos‘(π / 5)))
2	1	goldratmolem2 48136	. . . 4 ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))
3	2	oveq1i 7450	. . 3 ⊢ ( -1 · 2) = (((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2))) · 2)
4		2cn 12442	. . . 4 ⊢ 2 ∈ ℂ
5	4	mulm1i 11785	. . 3 ⊢ ( -1 · 2) = -2
6	1	goldrarr 48131	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ
7	6	recni 11347	. . . . . . . 8 ⊢ 𝐹 ∈ ℂ
8		5nn0 12650	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
9		expcl 14246	. . . . . . . 8 ⊢ ((𝐹 ∈ ℂ ∧ 5 ∈ ℕ₀) → (𝐹↑5) ∈ ℂ)
10	7, 8, 9	mp2an 705	. . . . . . 7 ⊢ (𝐹↑5) ∈ ℂ
11		2ne0 12473	. . . . . . 7 ⊢ 2 ≠ 0
12	10, 4, 11	divcli 12083	. . . . . 6 ⊢ ((𝐹↑5) / 2) ∈ ℂ
13		5cn 12455	. . . . . . 7 ⊢ 5 ∈ ℂ
14		3nn0 12648	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
15		expcl 14246	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐹↑3) ∈ ℂ)
16	7, 14, 15	mp2an 705	. . . . . . . 8 ⊢ (𝐹↑3) ∈ ℂ
17	16, 4, 11	divcli 12083	. . . . . . 7 ⊢ ((𝐹↑3) / 2) ∈ ℂ
18	13, 17	mulcli 11340	. . . . . 6 ⊢ (5 · ((𝐹↑3) / 2)) ∈ ℂ
19	12, 18	subcli 11658	. . . . 5 ⊢ (((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) ∈ ℂ
20	7, 4, 11	divcli 12083	. . . . . 6 ⊢ (𝐹 / 2) ∈ ℂ
21	13, 20	mulcli 11340	. . . . 5 ⊢ (5 · (𝐹 / 2)) ∈ ℂ
22	19, 21, 4	adddiri 11346	. . . 4 ⊢ (((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2))) · 2) = (((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) · 2) + ((5 · (𝐹 / 2)) · 2))
23	12, 18, 4	subdiri 11790	. . . . . 6 ⊢ ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) · 2) = ((((𝐹↑5) / 2) · 2) − ((5 · ((𝐹↑3) / 2)) · 2))
24	10, 4, 11	divcan1i 12085	. . . . . . 7 ⊢ (((𝐹↑5) / 2) · 2) = (𝐹↑5)
25	13, 17, 4	mulassi 11344	. . . . . . . 8 ⊢ ((5 · ((𝐹↑3) / 2)) · 2) = (5 · (((𝐹↑3) / 2) · 2))
26	16, 4, 11	divcan1i 12085	. . . . . . . . 9 ⊢ (((𝐹↑3) / 2) · 2) = (𝐹↑3)
27	26	oveq2i 7451	. . . . . . . 8 ⊢ (5 · (((𝐹↑3) / 2) · 2)) = (5 · (𝐹↑3))
28	25, 27	eqtri 2786	. . . . . . 7 ⊢ ((5 · ((𝐹↑3) / 2)) · 2) = (5 · (𝐹↑3))
29	24, 28	oveq12i 7452	. . . . . 6 ⊢ ((((𝐹↑5) / 2) · 2) − ((5 · ((𝐹↑3) / 2)) · 2)) = ((𝐹↑5) − (5 · (𝐹↑3)))
30	23, 29	eqtri 2786	. . . . 5 ⊢ ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) · 2) = ((𝐹↑5) − (5 · (𝐹↑3)))
31	13, 20, 4	mulassi 11344	. . . . . 6 ⊢ ((5 · (𝐹 / 2)) · 2) = (5 · ((𝐹 / 2) · 2))
32	7, 4, 11	divcan1i 12085	. . . . . . 7 ⊢ ((𝐹 / 2) · 2) = 𝐹
33	32	oveq2i 7451	. . . . . 6 ⊢ (5 · ((𝐹 / 2) · 2)) = (5 · 𝐹)
34	31, 33	eqtri 2786	. . . . 5 ⊢ ((5 · (𝐹 / 2)) · 2) = (5 · 𝐹)
35	30, 34	oveq12i 7452	. . . 4 ⊢ (((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) · 2) + ((5 · (𝐹 / 2)) · 2)) = (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹))
36	22, 35	eqtri 2786	. . 3 ⊢ (((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2))) · 2) = (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹))
37	3, 5, 36	3eqtr3ri 2795	. 2 ⊢ (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) = -2
38	13, 16	mulcli 11340	. . . . 5 ⊢ (5 · (𝐹↑3)) ∈ ℂ
39	10, 38	subcli 11658	. . . 4 ⊢ ((𝐹↑5) − (5 · (𝐹↑3))) ∈ ℂ
40	13, 7	mulcli 11340	. . . 4 ⊢ (5 · 𝐹) ∈ ℂ
41	39, 40	addcli 11339	. . 3 ⊢ (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) ∈ ℂ
42		addeq0 11763	. . 3 ⊢ (((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) ∈ ℂ ∧ 2 ∈ ℂ) → (((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0 ↔ (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) = -2))
43	41, 4, 42	mp2an 705	. 2 ⊢ (((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0 ↔ (((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) = -2)
44	37, 43	mpbir 234	1 ⊢ ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1570 ∈ wcel 2147 ‘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 5c5 12424 ℕ₀cn0 12630 ↑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: goldratmolem4 48138

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