ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 13912

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 11736, 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 7927	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11703	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addid2 8086	. . . . . . . . . . . . . . 15
6	5	oveq1d 5884	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 7941	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7927	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1017	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8296	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 999	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5885	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2219	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8121	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2228	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5515	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 13904	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11719	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2218	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8142	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2212	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5885	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8146	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11703	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1017	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11703	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8265	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2210	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5884	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5885	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8146	. . . . . . . 8 $/\$
37	36	3ad2ant1 1018	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11703	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8258	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 7967	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 8979	. . . . . . 7
42		2ap0 9001	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8643	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8285	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5884	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2212	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2210	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11736	. . . 4 $/\$
52	51	3ad2ant1 1018	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11736	. . . 4 $/\$
54	53	3ad2ant2 1019	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5887	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8296	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5515	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1017	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1177	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1017	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 7932	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5876	. . . . . . . . . . 11
68	67	3ad2ant3 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 109	. . . . . . . . . . . . . 14 $/\$
70		simpr 110	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1177	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8189	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5885	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 997	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8108	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7927	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7927	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1017	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8084	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2214	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 13875	. . . . . . . . . . 11
92		addcom 8084	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1017	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2210	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5515	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 13906	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1017	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2210	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5887	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8146	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11703	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8265	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1017	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5884	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11736	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1017	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11719	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8206	. . . . . . . 8 $/\$
116	115	fveq2d 5515	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2212	. . . . . 6 $/\$
118	117	3ad2ant1 1018	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5884	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5884	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2220	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2220	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 978

wceq 1353

wcel 2148 class class class wbr 4000

cfv 5212 (class class class)co 5869

cc 7800

cc0 7802

caddc 7805

cmul 7807

cmin 8118

cneg 8119 # cap 8528

cdiv 8618

c2 8959

csin 11636

ccos 11637

cpi 11639

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 ax-pre-suploc 7923 ax-addf 7924 ax-mulf 7925

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-of 6077 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-map 6644 df-pm 6645 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-5 8970 df-6 8971 df-7 8972 df-8 8973 df-9 8974 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-ioo 9879 df-ioc 9880 df-ico 9881 df-icc 9882 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-fac 10690 df-bc 10712 df-ihash 10740 df-shft 10808 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346 df-ef 11640 df-sin 11642 df-cos 11643 df-pi 11645 df-rest 12638 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208 df-ntr 13263 df-cn 13355 df-cnp 13356 df-tx 13420 df-cncf 13725 df-limced 13792 df-dvap 13793

This theorem is referenced by: (None)

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