ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 13539

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 11707, 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 7899	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1014	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11674	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8221	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addid2 8058	. . . . . . . . . . . . . . 15
6	5	oveq1d 5868	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 7913	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7899	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1012	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8268	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 994	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5869	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2212	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8093	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2221	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5500	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 13531	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11690	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2211	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8114	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2205	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5869	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8118	. . . . . . . . . 10 $/\$
28	27	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11674	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1012	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8237	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2203	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5868	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5869	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8118	. . . . . . . 8 $/\$
37	36	3ad2ant1 1013	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11674	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8230	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 7939	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 8949	. . . . . . 7
42		2ap0 8971	. . . . . . 7 #
43	41, 42	pm3.2i 270	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8614	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1233	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8257	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5868	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2205	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2203	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11707	. . . 4 $/\$
52	51	3ad2ant1 1013	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11707	. . . 4 $/\$
54	53	3ad2ant2 1014	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5871	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 525	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 524	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 526	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8268	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5500	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1012	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1172	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1012	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 7904	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5860	. . . . . . . . . . 11
68	67	3ad2ant3 1015	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 108	. . . . . . . . . . . . . 14 $/\$
70		simpr 109	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1172	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1014	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8161	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5869	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 992	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1014	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8080	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 409	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7899	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 506	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7899	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 507	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1012	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8056	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2207	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2212	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 13502	. . . . . . . . . . 11
92		addcom 8056	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 412	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1012	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2203	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5500	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 13533	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1012	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2203	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5871	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8118	. . . . . . . . . 10 $/\$
103	102	ancoms 266	. . . . . . . . 9 $/\$
104	103	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11674	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8237	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1012	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2203	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5868	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11707	. . . . 5 $/\$
111	84, 82, 110	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1012	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11690	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8178	. . . . . . . 8 $/\$
116	115	fveq2d 5500	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2205	. . . . . 6 $/\$
118	117	3ad2ant1 1013	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5868	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5868	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2213	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2213	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 973

wceq 1348

wcel 2141 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc 7772

cc0 7774

caddc 7777

cmul 7779

cmin 8090

cneg 8091 # cap 8500

cdiv 8589

c2 8929

csin 11607

ccos 11608

cpi 11610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 ax-pre-suploc 7895 ax-addf 7896 ax-mulf 7897

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-map 6628 df-pm 6629 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-ioo 9849 df-ioc 9850 df-ico 9851 df-icc 9852 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-sin 11613 df-cos 11614 df-pi 11616 df-rest 12581 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-topon 12803 df-bases 12835 df-ntr 12890 df-cn 12982 df-cnp 12983 df-tx 13047 df-cncf 13352 df-limced 13419 df-dvap 13420

This theorem is referenced by: (None)

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