ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15411

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 12170, 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 8085	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1022	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12137	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8246	. . . . . . . . . . . . . . 15
6	5	oveq1d 5982	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8100	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8085	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1020	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8456	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1002	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5983	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2249	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8281	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2258	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5603	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15403	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12153	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2248	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2242	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5983	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8306	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12137	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12137	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8425	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2240	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5982	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5983	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8306	. . . . . . . 8 $/\$
37	36	3ad2ant1 1021	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12137	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8418	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8127	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9142	. . . . . . 7
42		2ap0 9164	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8805	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1250	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8445	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5982	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2242	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2240	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12170	. . . 4 $/\$
52	51	3ad2ant1 1021	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12170	. . . 4 $/\$
54	53	3ad2ant2 1022	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5985	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8456	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5603	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1020	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1180	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1020	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8090	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5974	. . . . . . . . . . 11
68	67	3ad2ant3 1023	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1180	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1022	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8349	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5983	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1000	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1022	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8268	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8085	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8085	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1020	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8244	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2244	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2249	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15374	. . . . . . . . . . 11
92		addcom 8244	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1020	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2240	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5603	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15405	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2240	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5985	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8306	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12137	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8425	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2240	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5982	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12170	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1020	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12153	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8366	. . . . . . . 8 $/\$
116	115	fveq2d 5603	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2242	. . . . . 6 $/\$
118	117	3ad2ant1 1021	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5982	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5982	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2250	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2250	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 981

wceq 1373

wcel 2178 class class class wbr 4059

cfv 5290 (class class class)co 5967

cc 7958

cc0 7960

caddc 7963

cmul 7965

cmin 8278

cneg 8279 # cap 8689

cdiv 8780

c2 9122

csin 12070

ccos 12071

cpi 12073

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080 ax-pre-suploc 8081 ax-addf 8082 ax-mulf 8083

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-disj 4036 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-of 6181 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-oadd 6529 df-er 6643 df-map 6760 df-pm 6761 df-en 6851 df-dom 6852 df-fin 6853 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-5 9133 df-6 9134 df-7 9135 df-8 9136 df-9 9137 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-xneg 9929 df-xadd 9930 df-ioo 10049 df-ioc 10050 df-ico 10051 df-icc 10052 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-fac 10908 df-bc 10930 df-ihash 10958 df-shft 11241 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-sumdc 11780 df-ef 12074 df-sin 12076 df-cos 12077 df-pi 12079 df-rest 13188 df-topgen 13207 df-psmet 14420 df-xmet 14421 df-met 14422 df-bl 14423 df-mopn 14424 df-top 14585 df-topon 14598 df-bases 14630 df-ntr 14683 df-cn 14775 df-cnp 14776 df-tx 14840 df-cncf 15158 df-limced 15243 df-dvap 15244

This theorem is referenced by: (None)

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