ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15267

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 12026, 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 8049	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11993	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8373	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8210	. . . . . . . . . . . . . . 15
6	5	oveq1d 5958	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8064	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8049	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1019	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8420	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1001	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5959	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2246	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8245	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2255	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5579	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15259	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12009	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2245	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8266	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2239	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5959	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8270	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11993	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11993	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8389	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2237	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5958	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5959	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8270	. . . . . . . 8 $/\$
37	36	3ad2ant1 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11993	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8382	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8091	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9106	. . . . . . 7
42		2ap0 9128	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8769	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8409	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5958	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2239	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2237	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12026	. . . 4 $/\$
52	51	3ad2ant1 1020	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12026	. . . 4 $/\$
54	53	3ad2ant2 1021	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5961	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8420	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5579	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1019	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1179	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1019	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8054	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5950	. . . . . . . . . . 11
68	67	3ad2ant3 1022	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1179	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8313	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5959	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 999	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8232	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8049	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8049	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8208	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2241	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2246	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15230	. . . . . . . . . . 11
92		addcom 8208	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2237	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5579	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15261	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2237	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5961	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8270	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11993	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8389	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2237	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5958	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12026	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1019	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12009	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8330	. . . . . . . 8 $/\$
116	115	fveq2d 5579	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2239	. . . . . 6 $/\$
118	117	3ad2ant1 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5958	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5958	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2247	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2247	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 980

wceq 1372

wcel 2175 class class class wbr 4043

cfv 5270 (class class class)co 5943

cc 7922

cc0 7924

caddc 7927

cmul 7929

cmin 8242

cneg 8243 # cap 8653

cdiv 8744

c2 9086

csin 11926

ccos 11927

cpi 11929

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044 ax-pre-suploc 8045 ax-addf 8046 ax-mulf 8047

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-disj 4021 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-of 6157 df-1st 6225 df-2nd 6226 df-recs 6390 df-irdg 6455 df-frec 6476 df-1o 6501 df-oadd 6505 df-er 6619 df-map 6736 df-pm 6737 df-en 6827 df-dom 6828 df-fin 6829 df-sup 7085 df-inf 7086 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-5 9097 df-6 9098 df-7 9099 df-8 9100 df-9 9101 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-xneg 9893 df-xadd 9894 df-ioo 10013 df-ioc 10014 df-ico 10015 df-icc 10016 df-fz 10130 df-fzo 10264 df-seqfrec 10591 df-exp 10682 df-fac 10869 df-bc 10891 df-ihash 10919 df-shft 11097 df-cj 11124 df-re 11125 df-im 11126 df-rsqrt 11280 df-abs 11281 df-clim 11561 df-sumdc 11636 df-ef 11930 df-sin 11932 df-cos 11933 df-pi 11935 df-rest 13044 df-topgen 13063 df-psmet 14276 df-xmet 14277 df-met 14278 df-bl 14279 df-mopn 14280 df-top 14441 df-topon 14454 df-bases 14486 df-ntr 14539 df-cn 14631 df-cnp 14632 df-tx 14696 df-cncf 15014 df-limced 15099 df-dvap 15100

This theorem is referenced by: (None)

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