ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 12953

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 11487, 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 7769	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1004	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11454	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8088	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addid2 7925	. . . . . . . . . . . . . . 15
6	5	oveq1d 5797	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 7783	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7769	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1002	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8135	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 984	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5798	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2182	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 7960	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2191	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5433	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 12945	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11470	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2181	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 7981	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2175	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5798	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 7985	. . . . . . . . . 10 $/\$
28	27	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11454	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1002	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11454	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8104	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2173	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5797	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5798	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 7985	. . . . . . . 8 $/\$
37	36	3ad2ant1 1003	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11454	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8097	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 7809	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 8815	. . . . . . 7
42		2ap0 8837	. . . . . . 7 #
43	41, 42	pm3.2i 270	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8481	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1217	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8124	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5797	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2175	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2173	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11487	. . . 4 $/\$
52	51	3ad2ant1 1003	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11487	. . . 4 $/\$
54	53	3ad2ant2 1004	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5800	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 520	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 519	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8135	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5433	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1002	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1162	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 7774	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5789	. . . . . . . . . . 11
68	67	3ad2ant3 1005	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 108	. . . . . . . . . . . . . 14 $/\$
70		simpr 109	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1162	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1004	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8028	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5798	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 982	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1004	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 7947	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 409	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7769	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 501	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7769	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 502	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1002	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 7923	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2177	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2182	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 12916	. . . . . . . . . . 11
92		addcom 7923	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1002	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2173	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5433	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 12947	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1002	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2173	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5800	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 7985	. . . . . . . . . 10 $/\$
103	102	ancoms 266	. . . . . . . . 9 $/\$
104	103	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11454	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8104	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1002	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2173	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5797	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11487	. . . . 5 $/\$
111	84, 82, 110	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1002	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11470	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8045	. . . . . . . 8 $/\$
116	115	fveq2d 5433	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2175	. . . . . 6 $/\$
118	117	3ad2ant1 1003	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5797	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5797	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2183	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2183	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 963

wceq 1332

wcel 1481 class class class wbr 3937

cfv 5131 (class class class)co 5782

cc 7642

cc0 7644

caddc 7647

cmul 7649

cmin 7957

cneg 7958 # cap 8367

cdiv 8456

c2 8795

csin 11387

ccos 11388

cpi 11390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834

This theorem is referenced by: (None)

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