ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15689

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 12430, 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 8252	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1046	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12397	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8575	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8412	. . . . . . . . . . . . . . 15
6	5	oveq1d 6065	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8267	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8252	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1044	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8622	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1026	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6066	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2274	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8447	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2283	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5674	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15681	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12413	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2273	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8468	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2267	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6066	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8472	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12397	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1044	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12397	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8591	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2265	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6065	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6066	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8472	. . . . . . . 8 $/\$
37	36	3ad2ant1 1045	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12397	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8584	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8293	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9308	. . . . . . 7
42		2ap0 9330	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8971	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1274	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8611	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6065	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2267	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2265	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12430	. . . 4 $/\$
52	51	3ad2ant1 1045	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12430	. . . 4 $/\$
54	53	3ad2ant2 1046	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6068	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 529	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 531	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8622	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5674	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1044	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1204	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1044	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8257	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6057	. . . . . . . . . . 11
68	67	3ad2ant3 1047	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1204	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1046	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8515	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6066	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1024	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1046	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8434	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8252	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8252	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1044	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8410	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2269	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15652	. . . . . . . . . . 11
92		addcom 8410	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1044	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2265	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15683	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1044	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2265	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6068	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8472	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12397	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8591	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1044	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2265	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6065	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12430	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1044	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12413	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8532	. . . . . . . 8 $/\$
116	115	fveq2d 5674	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2267	. . . . . 6 $/\$
118	117	3ad2ant1 1045	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6065	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6065	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2275	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2275	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1005

wceq 1398

wcel 2203 class class class wbr 4109

cfv 5352 (class class class)co 6050

cc 8125

cc0 8127

caddc 8130

cmul 8132

cmin 8444

cneg 8445 # cap 8855

cdiv 8946

c2 9288

csin 12330

ccos 12331

cpi 12333

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247 ax-pre-suploc 8248 ax-addf 8249 ax-mulf 8250

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-disj 4086 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-of 6266 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-oadd 6651 df-er 6767 df-map 6884 df-pm 6885 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-5 9299 df-6 9300 df-7 9301 df-8 9302 df-9 9303 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-xneg 10105 df-xadd 10106 df-ioo 10225 df-ioc 10226 df-ico 10227 df-icc 10228 df-fz 10343 df-fzo 10477 df-seqfrec 10810 df-exp 10901 df-fac 11088 df-bc 11110 df-ihash 11139 df-shft 11500 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039 df-ef 12334 df-sin 12336 df-cos 12337 df-pi 12339 df-rest 13454 df-topgen 13473 df-psmet 14691 df-xmet 14692 df-met 14693 df-bl 14694 df-mopn 14695 df-top 14863 df-topon 14876 df-bases 14908 df-ntr 14961 df-cn 15053 df-cnp 15054 df-tx 15118 df-cncf 15436 df-limced 15521 df-dvap 15522

This theorem is referenced by: (None)

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