ptolemy - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ptolemy		Unicode version

Theorem ptolemy 15547

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 12304, 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 8156	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1045	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12271	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8480	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8317	. . . . . . . . . . . . . . 15
6	5	oveq1d 6032	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8171	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8156	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1043	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8527	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1025	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6033	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2273	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8352	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15539	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12287	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2272	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8373	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6033	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8377	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12271	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1043	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12271	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8496	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6032	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6033	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8377	. . . . . . . 8 $/\$
37	36	3ad2ant1 1044	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12271	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8489	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8198	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9213	. . . . . . 7
42		2ap0 9235	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8876	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8516	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6032	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2266	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2264	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12304	. . . 4 $/\$
52	51	3ad2ant1 1044	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12304	. . . 4 $/\$
54	53	3ad2ant2 1045	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6035	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 529	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8527	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5643	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1043	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1203	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1043	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8161	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6024	. . . . . . . . . . 11
68	67	3ad2ant3 1046	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1203	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1045	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8420	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6033	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1023	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1045	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8339	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8156	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8156	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8315	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2268	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2273	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15510	. . . . . . . . . . 11
92		addcom 8315	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1043	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2264	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5643	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15541	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1043	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2264	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6035	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8377	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12271	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8496	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1043	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2264	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6032	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12304	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1043	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12287	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8437	. . . . . . . 8 $/\$
116	115	fveq2d 5643	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2266	. . . . . 6 $/\$
118	117	3ad2ant1 1044	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6032	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6032	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2274	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2274	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1004

wceq 1397

wcel 2202 class class class wbr 4088

cfv 5326 (class class class)co 6017

cc 8029

cc0 8031

caddc 8034

cmul 8036

cmin 8349

cneg 8350 # cap 8760

cdiv 8851

c2 9193

csin 12204

ccos 12205

cpi 12207

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151 ax-pre-suploc 8152 ax-addf 8153 ax-mulf 8154

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-of 6234 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-oadd 6585 df-er 6701 df-map 6818 df-pm 6819 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-9 9208 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-ioo 10126 df-ioc 10127 df-ico 10128 df-icc 10129 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-exp 10800 df-fac 10987 df-bc 11009 df-ihash 11037 df-shft 11375 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-sumdc 11914 df-ef 12208 df-sin 12210 df-cos 12211 df-pi 12213 df-rest 13323 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-met 14558 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766 df-ntr 14819 df-cn 14911 df-cnp 14912 df-tx 14976 df-cncf 15294 df-limced 15379 df-dvap 15380

This theorem is referenced by: (None)

W3C validator