ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15000

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 11890, 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 7999	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11857	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8323	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8160	. . . . . . . . . . . . . . 15
6	5	oveq1d 5934	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8014	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7999	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1019	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8370	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1001	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5935	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2235	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8195	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2244	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5559	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 14992	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11873	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2234	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8216	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2228	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5935	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8220	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11857	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11857	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8339	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2226	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5934	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5935	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8220	. . . . . . . 8 $/\$
37	36	3ad2ant1 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11857	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8332	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8041	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9055	. . . . . . 7
42		2ap0 9077	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8718	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8359	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5934	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2228	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2226	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11890	. . . 4 $/\$
52	51	3ad2ant1 1020	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11890	. . . 4 $/\$
54	53	3ad2ant2 1021	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5937	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8370	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5559	. . . . . . 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 8004	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5926	. . . . . . . . . . 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 8263	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5935	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 999	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8182	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7999	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7999	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8158	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2230	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2235	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 14963	. . . . . . . . . . 11
92		addcom 8158	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5559	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 14994	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2226	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5937	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8220	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11857	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8339	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5934	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11890	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1019	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11873	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8280	. . . . . . . 8 $/\$
116	115	fveq2d 5559	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2228	. . . . . 6 $/\$
118	117	3ad2ant1 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5934	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5934	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2236	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2236	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1364

wcel 2164 class class class wbr 4030

cfv 5255 (class class class)co 5919

cc 7872

cc0 7874

caddc 7877

cmul 7879

cmin 8192

cneg 8193 # cap 8602

cdiv 8693

c2 9035

csin 11790

ccos 11791

cpi 11793

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994 ax-pre-suploc 7995 ax-addf 7996 ax-mulf 7997

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-disj 4008 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-of 6132 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-map 6706 df-pm 6707 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-ioo 9961 df-ioc 9962 df-ico 9963 df-icc 9964 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-fac 10800 df-bc 10822 df-ihash 10850 df-shft 10962 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500 df-ef 11794 df-sin 11796 df-cos 11797 df-pi 11799 df-rest 12855 df-topgen 12874 df-psmet 14042 df-xmet 14043 df-met 14044 df-bl 14045 df-mopn 14046 df-top 14177 df-topon 14190 df-bases 14222 df-ntr 14275 df-cn 14367 df-cnp 14368 df-tx 14432 df-cncf 14750 df-limced 14835 df-dvap 14836

This theorem is referenced by: (None)

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