ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 16051

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 12536, 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 8310	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1050	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 12503	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8635	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8472	. . . . . . . . . . . . . . 15
6	5	oveq1d 6103	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8325	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 8310	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1048	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8682	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1030	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 6104	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2280	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8507	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2289	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5702	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 16043	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 12519	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2279	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8528	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6104	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8532	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 12503	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1048	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 12503	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8651	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2271	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 6103	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 6104	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8532	. . . . . . . 8 $/\$
37	36	3ad2ant1 1049	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 12503	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8644	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8351	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9384	. . . . . . 7
42		2ap0 9406	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 9035	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1278	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8671	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 6103	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2273	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2271	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 12536	. . . 4 $/\$
52	51	3ad2ant1 1049	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 12536	. . . 4 $/\$
54	53	3ad2ant2 1050	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 6106	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 533	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 531	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 535	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8682	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5702	. . . . . . 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 8315	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 6095	. . . . . . . . . . 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 8575	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 6104	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 1028	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1050	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8494	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 415	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 8310	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 513	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 8310	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 514	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1048	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8470	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2280	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 16014	. . . . . . . . . . 11
92		addcom 8470	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 418	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1048	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2271	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5702	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 16045	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1048	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2271	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 6106	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8532	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 12503	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8651	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1048	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2271	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 6103	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 12536	. . . . 5 $/\$
111	84, 82, 110	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1048	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 12519	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8592	. . . . . . . 8 $/\$
116	115	fveq2d 5702	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2273	. . . . . 6 $/\$
118	117	3ad2ant1 1049	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 6103	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 6103	. . 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 4131

cfv 5380 (class class class)co 6088

cc 8183

cc0 8185

caddc 8188

cmul 8190

cmin 8504

cneg 8505 # cap 8917

cdiv 9010

c2 9364

csin 12436

ccos 12437

cpi 12439

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-pre-suploc 8306 ax-addf 8307 ax-mulf 8308

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-ioo 10311 df-ioc 10312 df-ico 10313 df-icc 10314 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-shft 11602 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145 df-ef 12440 df-sin 12442 df-cos 12443 df-pi 12445 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15224 df-topon 15237 df-bases 15269 df-ntr 15322 df-cn 15414 df-cnp 15415 df-tx 15479 df-cncf 15797 df-limced 15882 df-dvap 15883

This theorem is used by: (None)

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