ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 14959

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 11887, 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 7997	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11854	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8321	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8158	. . . . . . . . . . . . . . 15
6	5	oveq1d 5933	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 8012	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7997	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1019	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8368	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 1001	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5934	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2235	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8193	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2244	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5558	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 14951	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11870	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2234	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8214	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2228	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5934	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8218	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11854	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11854	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8337	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2226	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5933	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5934	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8218	. . . . . . . 8 $/\$
37	36	3ad2ant1 1020	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11854	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8330	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 8039	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 9053	. . . . . . 7
42		2ap0 9075	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8716	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8357	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5933	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2228	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2226	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11887	. . . 4 $/\$
52	51	3ad2ant1 1020	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11887	. . . 4 $/\$
54	53	3ad2ant2 1021	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5936	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8368	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5558	. . . . . . 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 8002	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5925	. . . . . . . . . . 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 8261	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5934	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 999	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1021	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8180	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7997	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7997	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8156	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2230	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2235	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 14922	. . . . . . . . . . 11
92		addcom 8156	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2226	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5558	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 14953	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1019	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2226	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5936	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8218	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11854	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8337	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1019	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2226	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5933	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11887	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1019	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11870	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8278	. . . . . . . 8 $/\$
116	115	fveq2d 5558	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2228	. . . . . 6 $/\$
118	117	3ad2ant1 1020	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5933	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5933	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2236	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2236	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1364

wcel 2164 class class class wbr 4029

cfv 5254 (class class class)co 5918

cc 7870

cc0 7872

caddc 7875

cmul 7877

cmin 8190

cneg 8191 # cap 8600

cdiv 8691

c2 9033

csin 11787

ccos 11788

cpi 11790

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 ax-pre-suploc 7993 ax-addf 7994 ax-mulf 7995

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-disj 4007 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-map 6704 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-5 9044 df-6 9045 df-7 9046 df-8 9047 df-9 9048 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-xneg 9838 df-xadd 9839 df-ioo 9958 df-ioc 9959 df-ico 9960 df-icc 9961 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-bc 10819 df-ihash 10847 df-shft 10959 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 df-ef 11791 df-sin 11793 df-cos 11794 df-pi 11796 df-rest 12852 df-topgen 12871 df-psmet 14039 df-xmet 14040 df-met 14041 df-bl 14042 df-mopn 14043 df-top 14166 df-topon 14179 df-bases 14211 df-ntr 14264 df-cn 14356 df-cnp 14357 df-tx 14421 df-cncf 14726 df-limced 14810 df-dvap 14811

This theorem is referenced by: (None)

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