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Theorem metdstri 22654

Description: A generalization of the triangle inequality to the point-set distance function. Under the usual notation where the same symbol

denotes the point-point and point-set distance functions, this theorem would be written

. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Hypothesis**
Ref	Expression
metdscn.f	inf

**Assertion**
Ref	Expression
metdstri	$/\$ $/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

)

**Proof of Theorem metdstri**
Step	Hyp	Ref	Expression
1		simprr 796	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
2		simprl 794	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
3		rexsub 12064	. . . . . . . . . . . 12 $/\$
4	1, 2, 3	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5	4	oveq2d 6666	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpll 790	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
7	6	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
8		simprr 796	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
9	8	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
10		simprl 794	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
11	10	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
12	1, 2	resubcld 10458	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	2	leidd 10594	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
14		xmetsym 22152	. . . . . . . . . . . . . . 15 $/\$ $/\$
15	6, 10, 8, 14	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
16	15	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	eqcomd 2628	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
18	1	recnd 10068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
19	2	recnd 10068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	nncand 10397	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
21	13, 17, 20	3brtr4d 4685	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
22		blss2 22209	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
23	7, 9, 11, 12, 1, 21, 22	syl33anc 1341	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
24	5, 23	eqsstrd 3639	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
25	24	expr 643	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
26	6	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
27	8	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
28		metdscn.f	. . . . . . . . . . . . . . . . . 18 inf
29	28	metdsf 22651	. . . . . . . . . . . . . . . . 17 $/\$
30	29	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
31	30, 10	ffvelrnd 6360	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
32		elxrge0 12281	. . . . . . . . . . . . . . . 16 $/\$
33	32	simplbi 476	. . . . . . . . . . . . . . 15
34	31, 33	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
35	34	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
36		xmetcl 22136	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37	6, 10, 8, 36	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
38	37	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
39	38	xnegcld 12130	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
40	35, 39	xaddcld 12131	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
41	40	adantrr 753	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
42		pnfxr 10092	. . . . . . . . . . . 12
43	42	a1i 11	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
44		pnfge 11964	. . . . . . . . . . . 12
45	41, 44	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
46		ssbl 22228	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
47	26, 27, 41, 43, 45, 46	syl221anc 1337	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
48		simprr 796	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
49	48	oveq2d 6666	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
50	10	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
51		simprl 794	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
52		xblpnf 22201	. . . . . . . . . . . . . 14 $/\$ $/\$
53	26, 50, 52	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	27, 51, 53	mpbir2and 957	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
55		blpnfctr 22241	. . . . . . . . . . . 12 $/\$ $/\$
56	26, 50, 54, 55	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
57	49, 56	eqtr2d 2657	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
58	47, 57	sseqtrd 3641	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	expr 643	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
60	32	simprbi 480	. . . . . . . . . . . . 13
61	31, 60	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
62		ge0nemnf 12004	. . . . . . . . . . . 12 $/\$
63	34, 61, 62	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
64	34, 63	jca 554	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
65	64	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
66		xrnemnf 11951	. . . . . . . . 9 $/\$ $\/$
67	65, 66	sylib 208	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\/$
68	25, 59, 67	mpjaod 396	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
69		pnfnlt 11962	. . . . . . . . . . 11
70	34, 69	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$
71	70	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
72	37	xnegcld 12130	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
73	34, 72	xaddcld 12131	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74		xbln0 22219	. . . . . . . . . . . . 13 $/\$ $/\$
75	6, 8, 73, 74	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
76		xposdif 12092	. . . . . . . . . . . . 13 $/\$
77	37, 34, 76	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
78	75, 77	bitr4d 271	. . . . . . . . . . 11 $/\$ $/\$ $/\$
79		breq1 4656	. . . . . . . . . . 11
80	78, 79	sylan9bb 736	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
81	80	necon1bbid 2833	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
82	71, 81	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
83		0ss 3972	. . . . . . . 8
84	82, 83	syl6eqss 3655	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
85		xmetge0 22149	. . . . . . . . . . 11 $/\$ $/\$
86	6, 10, 8, 85	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$
87		ge0nemnf 12004	. . . . . . . . . 10 $/\$
88	37, 86, 87	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
89	37, 88	jca 554	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
90		xrnemnf 11951	. . . . . . . 8 $/\$ $\/$
91	89, 90	sylib 208	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
92	68, 84, 91	mpjaodan 827	. . . . . 6 $/\$ $/\$ $/\$
93		sslin 3839	. . . . . 6
94	92, 93	syl 17	. . . . 5 $/\$ $/\$ $/\$
95		xrleid 11983	. . . . . . 7
96	34, 95	syl 17	. . . . . 6 $/\$ $/\$ $/\$
97		simplr 792	. . . . . . 7 $/\$ $/\$ $/\$
98	28	metdsge 22652	. . . . . . 7 $/\$ $/\$ $/\$
99	6, 97, 10, 34, 98	syl31anc 1329	. . . . . 6 $/\$ $/\$ $/\$
100	96, 99	mpbid 222	. . . . 5 $/\$ $/\$ $/\$
101		sseq0 3975	. . . . 5 $/\$
102	94, 100, 101	syl2anc 693	. . . 4 $/\$ $/\$ $/\$
103	28	metdsge 22652	. . . . 5 $/\$ $/\$ $/\$
104	6, 97, 8, 73, 103	syl31anc 1329	. . . 4 $/\$ $/\$ $/\$
105	102, 104	mpbird 247	. . 3 $/\$ $/\$ $/\$
106	30, 8	ffvelrnd 6360	. . . . 5 $/\$ $/\$ $/\$
107		elxrge0 12281	. . . . . 6 $/\$
108	107	simplbi 476	. . . . 5
109	106, 108	syl 17	. . . 4 $/\$ $/\$ $/\$
110	107	simprbi 480	. . . . 5
111	106, 110	syl 17	. . . 4 $/\$ $/\$ $/\$
112		xlesubadd 12093	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113	34, 37, 109, 61, 88, 111, 112	syl33anc 1341	. . 3 $/\$ $/\$ $/\$
114	105, 113	mpbid 222	. 2 $/\$ $/\$ $/\$
115		xaddcom 12071	. . 3 $/\$
116	109, 37, 115	syl2anc 693	. 2 $/\$ $/\$ $/\$
117	114, 116	breqtrd 4679	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

cin 3573

wss 3574

c0 3915 class class class wbr 4653

cmpt 4729

crn 5115

wf 5884

cfv 5888 (class class class)co 6650 infcinf 8347

cr 9935

cc0 9936

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cle 10075

cmin 10266

cxne 11943

cxad 11944

cicc 12178

cxmt 19731

cbl 19733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-ec 7744 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-psmet 19738 df-xmet 19739 df-bl 19741

This theorem is referenced by: metdsle 22655 metdscnlem 22658 metnrmlem1 22662

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