ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15498

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 12255, 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 8124	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1043	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12222	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8448	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8285	. . . . . . . . . . . . . . 15
6	5	oveq1d 6016	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8139	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8124	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1041	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8495	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1023	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6017	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2271	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8320	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5631	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15490	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12238	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2270	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8341	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6017	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8345	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12222	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8464	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6016	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6017	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8345	. . . . . . . 8 $/\$
37	36	3ad2ant1 1042	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12222	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8457	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8166	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9181	. . . . . . 7
42		2ap0 9203	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8844	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8484	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6016	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2262	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12255	. . . 4 $/\$
52	51	3ad2ant1 1042	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12255	. . . 4 $/\$
54	53	3ad2ant2 1043	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6019	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8495	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5631	. . . . . . 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 8129	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6008	. . . . . . . . . . 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 8388	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6017	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8307	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8124	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8124	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1041	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8283	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2266	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2271	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15461	. . . . . . . . . . 11
92		addcom 8283	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5631	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15492	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2262	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6019	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8345	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12222	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8464	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1041	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6016	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12255	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1041	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12238	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8405	. . . . . . . 8 $/\$
116	115	fveq2d 5631	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2264	. . . . . 6 $/\$
118	117	3ad2ant1 1042	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6016	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6016	. . 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 6001

cc 7997

cc0 7999

caddc 8002

cmul 8004

cmin 8317

cneg 8318 # cap 8728

cdiv 8819

c2 9161

csin 12155

ccos 12156

cpi 12158

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 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119 ax-pre-suploc 8120 ax-addf 8121 ax-mulf 8122

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 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-of 6218 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-oadd 6566 df-er 6680 df-map 6797 df-pm 6798 df-en 6888 df-dom 6889 df-fin 6890 df-sup 7151 df-inf 7152 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-5 9172 df-6 9173 df-7 9174 df-8 9175 df-9 9176 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-xneg 9968 df-xadd 9969 df-ioo 10088 df-ioc 10089 df-ico 10090 df-icc 10091 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-exp 10761 df-fac 10948 df-bc 10970 df-ihash 10998 df-shft 11326 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-sumdc 11865 df-ef 12159 df-sin 12161 df-cos 12162 df-pi 12164 df-rest 13274 df-topgen 13293 df-psmet 14507 df-xmet 14508 df-met 14509 df-bl 14510 df-mopn 14511 df-top 14672 df-topon 14685 df-bases 14717 df-ntr 14770 df-cn 14862 df-cnp 14863 df-tx 14927 df-cncf 15245 df-limced 15330 df-dvap 15331

This theorem is referenced by: (None)

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