ptolemy - Metamath Proof Explorer

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Theorem ptolemy 20357

Description: Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12728, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. (Contributed by David A. Wheeler, 31-May-2015.)

**Assertion**
Ref	Expression
ptolemy	$/\$ $/\$ $/\$ $/\$

**Proof of Theorem ptolemy**
Step	Hyp	Ref	Expression
1		addcl 9028	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 979	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12687	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 9358	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addid2 9205	. . . . . . . . . . . . . . 15
6	5	oveq1d 6055	. . . . . . . . . . . . . 14
7	2, 6	syl 16	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cn 9040	. . . . . . . . . . . . . . 15
9	8	a1i 11	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
10		addcl 9028	. . . . . . . . . . . . . . . 16 $/\$
11	10	adantr 452	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
12	11	3adant3 977	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
13	9, 12, 2	pnpcan2d 9405	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
14		simp3 959	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
15	14	oveq2d 6056	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
16	7, 13, 15	3eqtr3rd 2445	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
17		df-neg 9250	. . . . . . . . . . . 12
18	16, 17	syl6eqr 2454	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
19	18	fveq2d 5691	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
20		cosmpi 20349	. . . . . . . . . . 11
21	2, 20	syl 16	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		cosneg 12703	. . . . . . . . . . 11
23	12, 22	syl 16	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
24	19, 21, 23	3eqtr3d 2444	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
25	24	negeqd 9256	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
26	4, 25	eqtr3d 2438	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
27	26	oveq2d 6056	. . . . . 6 $/\$ $/\$ $/\$ $/\$
28		subcl 9261	. . . . . . . . . 10 $/\$
29	28	adantl 453	. . . . . . . . 9 $/\$ $/\$ $/\$
30	29	coscld 12687	. . . . . . . 8 $/\$ $/\$ $/\$
31	30	3adant3 977	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	12	coscld 12687	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
33	31, 32	subnegd 9374	. . . . . 6 $/\$ $/\$ $/\$ $/\$
34	27, 33	eqtrd 2436	. . . . 5 $/\$ $/\$ $/\$ $/\$
35	34	oveq1d 6055	. . . 4 $/\$ $/\$ $/\$ $/\$
36	35	oveq2d 6056	. . 3 $/\$ $/\$ $/\$ $/\$
37		subcl 9261	. . . . . . . 8 $/\$
38	37	3ad2ant1 978	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	38	coscld 12687	. . . . . 6 $/\$ $/\$ $/\$ $/\$
40	39, 32	subcld 9367	. . . . 5 $/\$ $/\$ $/\$ $/\$
41	31, 32	addcld 9063	. . . . 5 $/\$ $/\$ $/\$ $/\$
42		2cn 10026	. . . . . . 7
43		2ne0 10039	. . . . . . 7
44	42, 43	pm3.2i 442	. . . . . 6 $/\$
45	44	a1i 11	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
46		divdir 9657	. . . . 5 $/\$ $/\$ $/\$
47	40, 41, 45, 46	syl3anc 1184	. . . 4 $/\$ $/\$ $/\$ $/\$
48	39, 32, 31	nppcan3d 9394	. . . . 5 $/\$ $/\$ $/\$ $/\$
49	48	oveq1d 6055	. . . 4 $/\$ $/\$ $/\$ $/\$
50	47, 49	eqtr3d 2438	. . 3 $/\$ $/\$ $/\$ $/\$
51	36, 50	eqtrd 2436	. 2 $/\$ $/\$ $/\$ $/\$
52		sinmul 12728	. . . 4 $/\$
53	52	3ad2ant1 978	. . 3 $/\$ $/\$ $/\$ $/\$
54		sinmul 12728	. . . 4 $/\$
55	54	3ad2ant2 979	. . 3 $/\$ $/\$ $/\$ $/\$
56	53, 55	oveq12d 6058	. 2 $/\$ $/\$ $/\$ $/\$
57		simplr 732	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simpll 731	. . . . . . . . 9 $/\$ $/\$ $/\$
59		simprl 733	. . . . . . . . 9 $/\$ $/\$ $/\$
60	57, 58, 59	pnpcan2d 9405	. . . . . . . 8 $/\$ $/\$ $/\$
61	60	fveq2d 5691	. . . . . . 7 $/\$ $/\$ $/\$
62	61	3adant3 977	. . . . . 6 $/\$ $/\$ $/\$ $/\$
63	1	adantl 453	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
64	11, 63, 29	3jca 1134	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
65	64	3adant3 977	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66		addass 9033	. . . . . . . . . . 11 $/\$ $/\$
67	65, 66	syl 16	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
68		oveq1 6047	. . . . . . . . . . 11
69	68	3ad2ant3 980	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
70		simpl 444	. . . . . . . . . . . . . 14 $/\$
71		simpr 448	. . . . . . . . . . . . . 14 $/\$
72	70, 71, 70	3jca 1134	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
73	72	3ad2ant2 979	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		ppncan 9299	. . . . . . . . . . . . 13 $/\$ $/\$
75	74	oveq2d 6056	. . . . . . . . . . . 12 $/\$ $/\$
76	73, 75	syl 16	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
77		simp1 957	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
78	70, 70	jca 519	. . . . . . . . . . . . 13 $/\$ $/\$
79	78	3ad2ant2 979	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
80		add4 9237	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
81	77, 79, 80	syl2anc 643	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
82		addcl 9028	. . . . . . . . . . . . . . 15 $/\$
83	82	ad2ant2r 728	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
84		addcl 9028	. . . . . . . . . . . . . . 15 $/\$
85	84	ad2ant2lr 729	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
86	83, 85	jca 519	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	86	3adant3 977	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
88		addcom 9208	. . . . . . . . . . . 12 $/\$
89	87, 88	syl 16	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
90	76, 81, 89	3eqtrd 2440	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
91	67, 69, 90	3eqtr3rd 2445	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
92		pire 20325	. . . . . . . . . . . 12
93	92	recni 9058	. . . . . . . . . . 11
94		addcom 9208	. . . . . . . . . . 11 $/\$
95	93, 29, 94	sylancr 645	. . . . . . . . . 10 $/\$ $/\$ $/\$
96	95	3adant3 977	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
97	91, 96	eqtrd 2436	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
98	97	fveq2d 5691	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
99		cosppi 20351	. . . . . . . . 9
100	29, 99	syl 16	. . . . . . . 8 $/\$ $/\$ $/\$
101	100	3adant3 977	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
102	98, 101	eqtrd 2436	. . . . . 6 $/\$ $/\$ $/\$ $/\$
103	62, 102	oveq12d 6058	. . . . 5 $/\$ $/\$ $/\$ $/\$
104		subcl 9261	. . . . . . . . . 10 $/\$
105	104	ancoms 440	. . . . . . . . 9 $/\$
106	105	adantr 452	. . . . . . . 8 $/\$ $/\$ $/\$
107	106	coscld 12687	. . . . . . 7 $/\$ $/\$ $/\$
108	107, 30	subnegd 9374	. . . . . 6 $/\$ $/\$ $/\$
109	108	3adant3 977	. . . . 5 $/\$ $/\$ $/\$ $/\$
110	103, 109	eqtrd 2436	. . . 4 $/\$ $/\$ $/\$ $/\$
111	110	oveq1d 6055	. . 3 $/\$ $/\$ $/\$ $/\$
112		sinmul 12728	. . . . 5 $/\$
113	85, 83, 112	syl2anc 643	. . . 4 $/\$ $/\$ $/\$
114	113	3adant3 977	. . 3 $/\$ $/\$ $/\$ $/\$
115		cosneg 12703	. . . . . . . 8
116	37, 115	syl 16	. . . . . . 7 $/\$
117		negsubdi2 9316	. . . . . . . 8 $/\$
118	117	fveq2d 5691	. . . . . . 7 $/\$
119	116, 118	eqtr3d 2438	. . . . . 6 $/\$
120	119	3ad2ant1 978	. . . . 5 $/\$ $/\$ $/\$ $/\$
121	120	oveq1d 6055	. . . 4 $/\$ $/\$ $/\$ $/\$
122	121	oveq1d 6055	. . 3 $/\$ $/\$ $/\$ $/\$
123	111, 114, 122	3eqtr4d 2446	. 2 $/\$ $/\$ $/\$ $/\$
124	51, 56, 123	3eqtr4d 2446	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wne 2567

cfv 5413 (class class class)co 6040

cc 8944

cc0 8946

caddc 8949

cmul 8951

cmin 9247

cneg 9248

cdiv 9633

c2 10005

csin 12621

ccos 12622

cpi 12624

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-inf2 7552 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 ax-pre-sup 9024 ax-addf 9025 ax-mulf 9026

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-pss 3296 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-tp 3782 df-op 3783 df-uni 3976 df-int 4011 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-tr 4263 df-eprel 4454 df-id 4458 df-po 4463 df-so 4464 df-fr 4501 df-se 4502 df-we 4503 df-ord 4544 df-on 4545 df-lim 4546 df-suc 4547 df-om 4805 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-isom 5422 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-of 6264 df-1st 6308 df-2nd 6309 df-riota 6508 df-recs 6592 df-rdg 6627 df-1o 6683 df-2o 6684 df-oadd 6687 df-er 6864 df-map 6979 df-pm 6980 df-ixp 7023 df-en 7069 df-dom 7070 df-sdom 7071 df-fin 7072 df-fi 7374 df-sup 7404 df-oi 7435 df-card 7782 df-cda 8004 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 df-div 9634 df-nn 9957 df-2 10014 df-3 10015 df-4 10016 df-5 10017 df-6 10018 df-7 10019 df-8 10020 df-9 10021 df-10 10022 df-n0 10178 df-z 10239 df-dec 10339 df-uz 10445 df-q 10531 df-rp 10569 df-xneg 10666 df-xadd 10667 df-xmul 10668 df-ioo 10876 df-ioc 10877 df-ico 10878 df-icc 10879 df-fz 11000 df-fzo 11091 df-fl 11157 df-seq 11279 df-exp 11338 df-fac 11522 df-bc 11549 df-hash 11574 df-shft 11837 df-cj 11859 df-re 11860 df-im 11861 df-sqr 11995 df-abs 11996 df-limsup 12220 df-clim 12237 df-rlim 12238 df-sum 12435 df-ef 12625 df-sin 12627 df-cos 12628 df-pi 12630 df-struct 13426 df-ndx 13427 df-slot 13428 df-base 13429 df-sets 13430 df-ress 13431 df-plusg 13497 df-mulr 13498 df-starv 13499 df-sca 13500 df-vsca 13501 df-tset 13503 df-ple 13504 df-ds 13506 df-unif 13507 df-hom 13508 df-cco 13509 df-rest 13605 df-topn 13606 df-topgen 13622 df-pt 13623 df-prds 13626 df-xrs 13681 df-0g 13682 df-gsum 13683 df-qtop 13688 df-imas 13689 df-xps 13691 df-mre 13766 df-mrc 13767 df-acs 13769 df-mnd 14645 df-submnd 14694 df-mulg 14770 df-cntz 15071 df-cmn 15369 df-psmet 16649 df-xmet 16650 df-met 16651 df-bl 16652 df-mopn 16653 df-fbas 16654 df-fg 16655 df-cnfld 16659 df-top 16918 df-bases 16920 df-topon 16921 df-topsp 16922 df-cld 17038 df-ntr 17039 df-cls 17040 df-nei 17117 df-lp 17155 df-perf 17156 df-cn 17245 df-cnp 17246 df-haus 17333 df-tx 17547 df-hmeo 17740 df-fil 17831 df-fm 17923 df-flim 17924 df-flf 17925 df-xms 18303 df-ms 18304 df-tms 18305 df-cncf 18861 df-limc 19706 df-dv 19707

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