ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 14284

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 11754, 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 7938	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11721	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8261	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addlid 8098	. . . . . . . . . . . . . . 15
6	5	oveq1d 5892	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 7952	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7938	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1017	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8308	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 999	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5893	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2219	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8133	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2228	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5521	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 14276	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11737	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2218	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8154	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2212	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5893	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8158	. . . . . . . . . 10 $/\$
28	27	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11721	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1017	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11721	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2210	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5892	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5893	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8158	. . . . . . . 8 $/\$
37	36	3ad2ant1 1018	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11721	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8270	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 7979	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 8992	. . . . . . 7
42		2ap0 9014	. . . . . . 7 #
43	41, 42	pm3.2i 272	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8656	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8297	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5892	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2212	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2210	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11754	. . . 4 $/\$
52	51	3ad2ant1 1018	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11754	. . . 4 $/\$
54	53	3ad2ant2 1019	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5895	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 528	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 527	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8308	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5521	. . . . . . 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 7943	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5884	. . . . . . . . . . 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 8201	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5893	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 997	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1019	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8120	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 411	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7938	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 509	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7938	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 510	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 306	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1017	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8096	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2214	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 14247	. . . . . . . . . . 11
92		addcom 8096	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 414	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1017	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2210	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5521	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 14278	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1017	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2210	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5895	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8158	. . . . . . . . . 10 $/\$
103	102	ancoms 268	. . . . . . . . 9 $/\$
104	103	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11721	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8277	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1017	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2210	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5892	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11754	. . . . 5 $/\$
111	84, 82, 110	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1017	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11737	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8218	. . . . . . . 8 $/\$
116	115	fveq2d 5521	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2212	. . . . . 6 $/\$
118	117	3ad2ant1 1018	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5892	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5892	. . 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 4005

cfv 5218 (class class class)co 5877

cc 7811

cc0 7813

caddc 7816

cmul 7818

cmin 8130

cneg 8131 # cap 8540

cdiv 8631

c2 8972

csin 11654

ccos 11655

cpi 11657

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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-pre-suploc 7934 ax-addf 7935 ax-mulf 7936

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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-map 6652 df-pm 6653 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-ioo 9894 df-ioc 9895 df-ico 9896 df-icc 9897 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-shft 10826 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-sin 11660 df-cos 11661 df-pi 11663 df-rest 12695 df-topgen 12714 df-psmet 13486 df-xmet 13487 df-met 13488 df-bl 13489 df-mopn 13490 df-top 13537 df-topon 13550 df-bases 13582 df-ntr 13635 df-cn 13727 df-cnp 13728 df-tx 13792 df-cncf 14097 df-limced 14164 df-dvap 14165

This theorem is referenced by: (None)

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