ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15492

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 12250, 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 8120	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1043	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12217	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8444	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8281	. . . . . . . . . . . . . . 15
6	5	oveq1d 6015	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8120	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1041	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8491	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1023	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6016	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2271	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8316	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5630	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15484	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12233	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2270	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8337	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6016	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8341	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12217	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12217	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8460	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6015	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6016	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8341	. . . . . . . 8 $/\$
37	36	3ad2ant1 1042	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12217	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8453	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8162	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9177	. . . . . . 7
42		2ap0 9199	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8840	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8480	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6015	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2262	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12250	. . . 4 $/\$
52	51	3ad2ant1 1042	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12250	. . . 4 $/\$
54	53	3ad2ant2 1043	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6018	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8491	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5630	. . . . . . 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 8125	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6007	. . . . . . . . . . 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 8384	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6016	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8303	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8120	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8120	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1041	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8279	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2266	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2271	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15455	. . . . . . . . . . 11
92		addcom 8279	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15486	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2262	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6018	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8341	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12217	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8460	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1041	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6015	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12250	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1041	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12233	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8401	. . . . . . . 8 $/\$
116	115	fveq2d 5630	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2264	. . . . . 6 $/\$
118	117	3ad2ant1 1042	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6015	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6015	. . 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 4082

cfv 5317 (class class class)co 6000

cc 7993

cc0 7995

caddc 7998

cmul 8000

cmin 8313

cneg 8314 # cap 8724

cdiv 8815

c2 9157

csin 12150

ccos 12151

cpi 12153

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 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115 ax-pre-suploc 8116 ax-addf 8117 ax-mulf 8118

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 3888 df-int 3923 df-iun 3966 df-disj 4059 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-of 6216 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-frec 6535 df-1o 6560 df-oadd 6564 df-er 6678 df-map 6795 df-pm 6796 df-en 6886 df-dom 6887 df-fin 6888 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-5 9168 df-6 9169 df-7 9170 df-8 9171 df-9 9172 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-xneg 9964 df-xadd 9965 df-ioo 10084 df-ioc 10085 df-ico 10086 df-icc 10087 df-fz 10201 df-fzo 10335 df-seqfrec 10665 df-exp 10756 df-fac 10943 df-bc 10965 df-ihash 10993 df-shft 11321 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-clim 11785 df-sumdc 11860 df-ef 12154 df-sin 12156 df-cos 12157 df-pi 12159 df-rest 13269 df-topgen 13288 df-psmet 14501 df-xmet 14502 df-met 14503 df-bl 14504 df-mopn 14505 df-top 14666 df-topon 14679 df-bases 14711 df-ntr 14764 df-cn 14856 df-cnp 14857 df-tx 14921 df-cncf 15239 df-limced 15324 df-dvap 15325

This theorem is referenced by: (None)

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