ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 15296

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 12055, 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 8050	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1022	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12022	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8374	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8211	. . . . . . . . . . . . . . 15
6	5	oveq1d 5959	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8065	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8050	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1020	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8421	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1002	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5960	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2247	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8246	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2256	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5580	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 15288	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12038	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2246	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2240	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5960	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8271	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12022	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12022	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8390	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2238	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5959	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5960	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8271	. . . . . . . 8 $/\$
37	36	3ad2ant1 1021	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12022	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8383	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8092	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9107	. . . . . . 7
42		2ap0 9129	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8770	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1250	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8410	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5959	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2240	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2238	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12055	. . . 4 $/\$
52	51	3ad2ant1 1021	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12055	. . . 4 $/\$
54	53	3ad2ant2 1022	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5962	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8421	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5580	. . . . . . 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 8055	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5951	. . . . . . . . . . 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 8314	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5960	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1000	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1022	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8233	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8050	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8050	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1020	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8209	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2242	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2247	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15259	. . . . . . . . . . 11
92		addcom 8209	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1020	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2238	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5580	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 15290	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2238	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5962	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8271	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12022	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8390	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2238	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5959	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12055	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1020	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12038	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8331	. . . . . . . 8 $/\$
116	115	fveq2d 5580	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2240	. . . . . 6 $/\$
118	117	3ad2ant1 1021	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5959	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5959	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2248	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2248	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1373

wcel 2176 class class class wbr 4044

cfv 5271 (class class class)co 5944

cc 7923

cc0 7925

caddc 7928

cmul 7930

cmin 8243

cneg 8244 # cap 8654

cdiv 8745

c2 9087

csin 11955

ccos 11956

cpi 11958

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045 ax-pre-suploc 8046 ax-addf 8047 ax-mulf 8048

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-disj 4022 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-of 6158 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-oadd 6506 df-er 6620 df-map 6737 df-pm 6738 df-en 6828 df-dom 6829 df-fin 6830 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-xneg 9894 df-xadd 9895 df-ioo 10014 df-ioc 10015 df-ico 10016 df-icc 10017 df-fz 10131 df-fzo 10265 df-seqfrec 10593 df-exp 10684 df-fac 10871 df-bc 10893 df-ihash 10921 df-shft 11126 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-sumdc 11665 df-ef 11959 df-sin 11961 df-cos 11962 df-pi 11964 df-rest 13073 df-topgen 13092 df-psmet 14305 df-xmet 14306 df-met 14307 df-bl 14308 df-mopn 14309 df-top 14470 df-topon 14483 df-bases 14515 df-ntr 14568 df-cn 14660 df-cnp 14661 df-tx 14725 df-cncf 15043 df-limced 15128 df-dvap 15129

This theorem is referenced by: (None)

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