ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15513

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 12270, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. This is Metamath 100 proof #95. (Contributed by David A. Wheeler, 31-May-2015.)

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

**Proof of Theorem ptolemy**
Step	Hyp	Ref	Expression
1		addcl 8135	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1043	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12237	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8459	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8296	. . . . . . . . . . . . . . 15
6	5	oveq1d 6022	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8150	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8135	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1041	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8506	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1023	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6023	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2271	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8331	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5633	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15505	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12253	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2270	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8352	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6023	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8356	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12237	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12237	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8475	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6022	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6023	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8356	. . . . . . . 8 $/\$
37	36	3ad2ant1 1042	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12237	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8468	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8177	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9192	. . . . . . 7
42		2ap0 9214	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8855	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8495	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6022	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2262	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12270	. . . 4 $/\$
52	51	3ad2ant1 1042	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12270	. . . 4 $/\$
54	53	3ad2ant2 1043	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6025	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8506	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5633	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1041	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1201	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1041	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8140	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6014	. . . . . . . . . . 11
68	67	3ad2ant3 1044	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1201	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8399	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6023	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8318	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8135	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8135	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1041	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8294	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2266	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2271	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15476	. . . . . . . . . . 11
92		addcom 8294	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5633	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15507	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2262	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6025	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8356	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12237	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8475	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1041	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6022	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12270	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1041	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12253	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8416	. . . . . . . 8 $/\$
116	115	fveq2d 5633	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2264	. . . . . 6 $/\$
118	117	3ad2ant1 1042	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6022	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6022	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2272	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2272	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1002

wceq 1395

wcel 2200 class class class wbr 4083

cfv 5318 (class class class)co 6007

cc 8008

cc0 8010

caddc 8013

cmul 8015

cmin 8328

cneg 8329 # cap 8739

cdiv 8830

c2 9172

csin 12170

ccos 12171

cpi 12173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130 ax-pre-suploc 8131 ax-addf 8132 ax-mulf 8133

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7162 df-inf 7163 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-5 9183 df-6 9184 df-7 9185 df-8 9186 df-9 9187 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-xneg 9980 df-xadd 9981 df-ioo 10100 df-ioc 10101 df-ico 10102 df-icc 10103 df-fz 10217 df-fzo 10351 df-seqfrec 10682 df-exp 10773 df-fac 10960 df-bc 10982 df-ihash 11010 df-shft 11341 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-clim 11805 df-sumdc 11880 df-ef 12174 df-sin 12176 df-cos 12177 df-pi 12179 df-rest 13289 df-topgen 13308 df-psmet 14522 df-xmet 14523 df-met 14524 df-bl 14525 df-mopn 14526 df-top 14687 df-topon 14700 df-bases 14732 df-ntr 14785 df-cn 14877 df-cnp 14878 df-tx 14942 df-cncf 15260 df-limced 15345 df-dvap 15346

This theorem is referenced by: (None)

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