ptolemy - Intuitionistic Logic Explorer

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Theorem ptolemy 13385

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 11685, 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 7878	. . . . . . . . . . 11 $/\$
2	1	3ad2ant2 1009	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3	2	coscld 11652	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
4	3	negnegd 8200	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		addid2 8037	. . . . . . . . . . . . . . 15
6	5	oveq1d 5857	. . . . . . . . . . . . . 14
7	2, 6	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
8		0cnd 7892	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
9		addcl 7878	. . . . . . . . . . . . . . . 16 $/\$
10	9	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
11	10	3adant3 1007	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
12	8, 11, 2	pnpcan2d 8247	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
13		simp3 989	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
14	13	oveq2d 5858	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
15	7, 12, 14	3eqtr3rd 2207	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
16		df-neg 8072	. . . . . . . . . . . 12
17	15, 16	eqtr4di 2217	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18	17	fveq2d 5490	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
19		cosmpi 13377	. . . . . . . . . . 11
20	2, 19	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
21		cosneg 11668	. . . . . . . . . . 11
22	11, 21	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
23	18, 20, 22	3eqtr3d 2206	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	23	negeqd 8093	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25	4, 24	eqtr3d 2200	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5858	. . . . . 6 $/\$ $/\$ $/\$ $/\$
27		subcl 8097	. . . . . . . . . 10 $/\$
28	27	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	coscld 11652	. . . . . . . 8 $/\$ $/\$ $/\$
30	29	3adant3 1007	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
31	11	coscld 11652	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
32	30, 31	subnegd 8216	. . . . . 6 $/\$ $/\$ $/\$ $/\$
33	26, 32	eqtrd 2198	. . . . 5 $/\$ $/\$ $/\$ $/\$
34	33	oveq1d 5857	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	oveq2d 5858	. . 3 $/\$ $/\$ $/\$ $/\$
36		subcl 8097	. . . . . . . 8 $/\$
37	36	3ad2ant1 1008	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
38	37	coscld 11652	. . . . . 6 $/\$ $/\$ $/\$ $/\$
39	38, 31	subcld 8209	. . . . 5 $/\$ $/\$ $/\$ $/\$
40	30, 31	addcld 7918	. . . . 5 $/\$ $/\$ $/\$ $/\$
41		2cn 8928	. . . . . . 7
42		2ap0 8950	. . . . . . 7 #
43	41, 42	pm3.2i 270	. . . . . 6 $/\$ #
44	43	a1i 9	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ #
45		divdirap 8593	. . . . 5 $/\$ $/\$ $/\$ #
46	39, 40, 44, 45	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$
47	38, 31, 30	nppcan3d 8236	. . . . 5 $/\$ $/\$ $/\$ $/\$
48	47	oveq1d 5857	. . . 4 $/\$ $/\$ $/\$ $/\$
49	46, 48	eqtr3d 2200	. . 3 $/\$ $/\$ $/\$ $/\$
50	35, 49	eqtrd 2198	. 2 $/\$ $/\$ $/\$ $/\$
51		sinmul 11685	. . . 4 $/\$
52	51	3ad2ant1 1008	. . 3 $/\$ $/\$ $/\$ $/\$
53		sinmul 11685	. . . 4 $/\$
54	53	3ad2ant2 1009	. . 3 $/\$ $/\$ $/\$ $/\$
55	52, 54	oveq12d 5860	. 2 $/\$ $/\$ $/\$ $/\$
56		simplr 520	. . . . . . . . 9 $/\$ $/\$ $/\$
57		simpll 519	. . . . . . . . 9 $/\$ $/\$ $/\$
58		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$
59	56, 57, 58	pnpcan2d 8247	. . . . . . . 8 $/\$ $/\$ $/\$
60	59	fveq2d 5490	. . . . . . 7 $/\$ $/\$ $/\$
61	60	3adant3 1007	. . . . . 6 $/\$ $/\$ $/\$ $/\$
62	1	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
63	10, 62, 28	3jca 1167	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	3adant3 1007	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		addass 7883	. . . . . . . . . . 11 $/\$ $/\$
66	64, 65	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
67		oveq1 5849	. . . . . . . . . . 11
68	67	3ad2ant3 1010	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
69		simpl 108	. . . . . . . . . . . . . 14 $/\$
70		simpr 109	. . . . . . . . . . . . . 14 $/\$
71	69, 70, 69	3jca 1167	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
72	71	3ad2ant2 1009	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		ppncan 8140	. . . . . . . . . . . . 13 $/\$ $/\$
74	73	oveq2d 5858	. . . . . . . . . . . 12 $/\$ $/\$
75	72, 74	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
76		simp1 987	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
77	69, 69	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$
78	77	3ad2ant2 1009	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
79		add4 8059	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
80	76, 78, 79	syl2anc 409	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81		addcl 7878	. . . . . . . . . . . . . . 15 $/\$
82	81	ad2ant2r 501	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
83		addcl 7878	. . . . . . . . . . . . . . 15 $/\$
84	83	ad2ant2lr 502	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
85	82, 84	jca 304	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
86	85	3adant3 1007	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
87		addcom 8035	. . . . . . . . . . . 12 $/\$
88	86, 87	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	75, 80, 88	3eqtrd 2202	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	66, 68, 89	3eqtr3rd 2207	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91		picn 13348	. . . . . . . . . . 11
92		addcom 8035	. . . . . . . . . . 11 $/\$
93	91, 28, 92	sylancr 411	. . . . . . . . . 10 $/\$ $/\$ $/\$
94	93	3adant3 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
95	90, 94	eqtrd 2198	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96	95	fveq2d 5490	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
97		cosppi 13379	. . . . . . . . 9
98	28, 97	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
99	98	3adant3 1007	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
100	96, 99	eqtrd 2198	. . . . . 6 $/\$ $/\$ $/\$ $/\$
101	61, 100	oveq12d 5860	. . . . 5 $/\$ $/\$ $/\$ $/\$
102		subcl 8097	. . . . . . . . . 10 $/\$
103	102	ancoms 266	. . . . . . . . 9 $/\$
104	103	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$
105	104	coscld 11652	. . . . . . 7 $/\$ $/\$ $/\$
106	105, 29	subnegd 8216	. . . . . 6 $/\$ $/\$ $/\$
107	106	3adant3 1007	. . . . 5 $/\$ $/\$ $/\$ $/\$
108	101, 107	eqtrd 2198	. . . 4 $/\$ $/\$ $/\$ $/\$
109	108	oveq1d 5857	. . 3 $/\$ $/\$ $/\$ $/\$
110		sinmul 11685	. . . . 5 $/\$
111	84, 82, 110	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
112	111	3adant3 1007	. . 3 $/\$ $/\$ $/\$ $/\$
113		cosneg 11668	. . . . . . . 8
114	36, 113	syl 14	. . . . . . 7 $/\$
115		negsubdi2 8157	. . . . . . . 8 $/\$
116	115	fveq2d 5490	. . . . . . 7 $/\$
117	114, 116	eqtr3d 2200	. . . . . 6 $/\$
118	117	3ad2ant1 1008	. . . . 5 $/\$ $/\$ $/\$ $/\$
119	118	oveq1d 5857	. . . 4 $/\$ $/\$ $/\$ $/\$
120	119	oveq1d 5857	. . 3 $/\$ $/\$ $/\$ $/\$
121	109, 112, 120	3eqtr4d 2208	. 2 $/\$ $/\$ $/\$ $/\$
122	50, 55, 121	3eqtr4d 2208	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 968

wceq 1343

wcel 2136 class class class wbr 3982

cfv 5188 (class class class)co 5842

cc 7751

cc0 7753

caddc 7756

cmul 7758

cmin 8069

cneg 8070 # cap 8479

cdiv 8568

c2 8908

csin 11585

ccos 11586

cpi 11588

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ioc 9829 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-sin 11591 df-cos 11592 df-pi 11594 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266

This theorem is referenced by: (None)

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