ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15538

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 12295, 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 8147	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1043	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12262	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8471	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8308	. . . . . . . . . . . . . . 15
6	5	oveq1d 6028	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8162	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8147	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1041	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8518	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1023	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6029	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2271	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8343	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5639	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15530	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12278	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2270	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8364	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6029	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8368	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12262	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12262	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8487	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6028	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6029	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8368	. . . . . . . 8 $/\$
37	36	3ad2ant1 1042	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12262	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8480	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8189	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9204	. . . . . . 7
42		2ap0 9226	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8867	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8507	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6028	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2262	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12295	. . . 4 $/\$
52	51	3ad2ant1 1042	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12295	. . . 4 $/\$
54	53	3ad2ant2 1043	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6031	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8518	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5639	. . . . . . 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 8152	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6020	. . . . . . . . . . 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 8411	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6029	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1043	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8330	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8147	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8147	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1041	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8306	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2266	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2271	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15501	. . . . . . . . . . 11
92		addcom 8306	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1041	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2262	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5639	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15532	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1041	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2262	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6031	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8368	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12262	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8487	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1041	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2262	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6028	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12295	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1041	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12278	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8428	. . . . . . . 8 $/\$
116	115	fveq2d 5639	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2264	. . . . . 6 $/\$
118	117	3ad2ant1 1042	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6028	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6028	. . 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 4086

cfv 5324 (class class class)co 6013

cc 8020

cc0 8022

caddc 8025

cmul 8027

cmin 8340

cneg 8341 # cap 8751

cdiv 8842

c2 9184

csin 12195

ccos 12196

cpi 12198

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142 ax-pre-suploc 8143 ax-addf 8144 ax-mulf 8145

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-disj 4063 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-map 6814 df-pm 6815 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7174 df-inf 7175 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-9 9199 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-xneg 9997 df-xadd 9998 df-ioo 10117 df-ioc 10118 df-ico 10119 df-icc 10120 df-fz 10234 df-fzo 10368 df-seqfrec 10700 df-exp 10791 df-fac 10978 df-bc 11000 df-ihash 11028 df-shft 11366 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-clim 11830 df-sumdc 11905 df-ef 12199 df-sin 12201 df-cos 12202 df-pi 12204 df-rest 13314 df-topgen 13333 df-psmet 14547 df-xmet 14548 df-met 14549 df-bl 14550 df-mopn 14551 df-top 14712 df-topon 14725 df-bases 14757 df-ntr 14810 df-cn 14902 df-cnp 14903 df-tx 14967 df-cncf 15285 df-limced 15370 df-dvap 15371

This theorem is referenced by: (None)

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