ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 16017

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 12530, 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 8305	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1050	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12497	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8630	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8467	. . . . . . . . . . . . . . 15
6	5	oveq1d 6100	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8320	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8305	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1048	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8677	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1030	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6101	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2280	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8502	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2289	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5699	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 16009	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12513	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2279	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8523	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6101	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8527	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12497	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1048	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12497	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8646	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2271	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6100	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6101	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8527	. . . . . . . 8 $/\$
37	36	3ad2ant1 1049	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12497	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8639	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8346	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9378	. . . . . . 7
42		2ap0 9400	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 9030	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1278	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8666	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6100	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2273	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2271	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12530	. . . 4 $/\$
52	51	3ad2ant1 1049	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12530	. . . 4 $/\$
54	53	3ad2ant2 1050	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6103	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 533	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 531	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 535	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8677	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5699	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1048	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1208	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1048	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 8310	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6092	. . . . . . . . . . 11
68	67	3ad2ant3 1051	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1208	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1050	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8570	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6101	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1028	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1050	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8489	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 415	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8305	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 513	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8305	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 514	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1048	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8465	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2280	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 15980	. . . . . . . . . . 11
92		addcom 8465	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 418	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1048	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5699	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 16011	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1048	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6103	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8527	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12497	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8646	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1048	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6100	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12530	. . . . 5 $/\$
111	84, 82, 110	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1048	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12513	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8587	. . . . . . . 8 $/\$
116	115	fveq2d 5699	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2273	. . . . . 6 $/\$
118	117	3ad2ant1 1049	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6100	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6100	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2281	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2281	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104 $/\$ w3a 1009

wceq 1402

wcel 2209 class class class wbr 4130

cfv 5377 (class class class)co 6085

cc 8178

cc0 8180

caddc 8183

cmul 8185

cmin 8499

cneg 8500 # cap 8912

cdiv 9005

c2 9358

csin 12430

ccos 12431

cpi 12433

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-pre-suploc 8301 ax-addf 8302 ax-mulf 8303

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-ioc 10306 df-ico 10307 df-icc 10308 df-fz 10423 df-fzo 10561 df-seqfrec 10900 df-exp 10991 df-fac 11180 df-bc 11202 df-ihash 11231 df-shft 11596 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-ef 12434 df-sin 12436 df-cos 12437 df-pi 12439 df-rest 13648 df-topgen 13667 df-psmet 14964 df-xmet 14965 df-met 14966 df-bl 14967 df-mopn 14968 df-top 15190 df-topon 15203 df-bases 15235 df-ntr 15288 df-cn 15380 df-cnp 15381 df-tx 15445 df-cncf 15763 df-limced 15848 df-dvap 15849

This theorem is used by: (None)

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