ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15060

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 11909, 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 8004	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11876	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8328	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8165	. . . . . . . . . . . . . . 15
6	5	oveq1d 5937	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8019	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8004	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1019	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8375	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1001	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5938	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2238	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8200	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2247	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5562	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15052	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11892	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2237	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8221	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2231	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5938	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8225	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11876	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11876	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8344	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2229	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5937	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5938	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8225	. . . . . . . 8 $/\$
37	36	3ad2ant1 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11876	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8337	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8046	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9061	. . . . . . 7
42		2ap0 9083	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8724	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8364	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5937	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2231	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2229	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11909	. . . 4 $/\$
52	51	3ad2ant1 1020	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11909	. . . 4 $/\$
54	53	3ad2ant2 1021	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5940	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8375	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5562	. . . . . . 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 8009	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5929	. . . . . . . . . . 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 8268	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5938	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 999	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8187	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8004	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8004	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8163	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2233	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2238	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15023	. . . . . . . . . . 11
92		addcom 8163	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2229	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5562	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15054	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2229	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5940	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8225	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11876	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8344	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2229	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5937	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11909	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1019	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11892	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8285	. . . . . . . 8 $/\$
116	115	fveq2d 5562	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2231	. . . . . 6 $/\$
118	117	3ad2ant1 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5937	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5937	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2239	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2239	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1364

wcel 2167 class class class wbr 4033

cfv 5258 (class class class)co 5922

cc 7877

cc0 7879

caddc 7882

cmul 7884

cmin 8197

cneg 8198 # cap 8608

cdiv 8699

c2 9041

csin 11809

ccos 11810

cpi 11812

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999 ax-pre-suploc 8000 ax-addf 8001 ax-mulf 8002

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-disj 4011 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-of 6135 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-map 6709 df-pm 6710 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-6 9053 df-7 9054 df-8 9055 df-9 9056 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-xneg 9847 df-xadd 9848 df-ioo 9967 df-ioc 9968 df-ico 9969 df-icc 9970 df-fz 10084 df-fzo 10218 df-seqfrec 10540 df-exp 10631 df-fac 10818 df-bc 10840 df-ihash 10868 df-shft 10980 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-sumdc 11519 df-ef 11813 df-sin 11815 df-cos 11816 df-pi 11818 df-rest 12912 df-topgen 12931 df-psmet 14099 df-xmet 14100 df-met 14101 df-bl 14102 df-mopn 14103 df-top 14234 df-topon 14247 df-bases 14279 df-ntr 14332 df-cn 14424 df-cnp 14425 df-tx 14489 df-cncf 14807 df-limced 14892 df-dvap 14893

This theorem is referenced by: (None)

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