Theorem dscmet 9196

Description: The discrete metric on any set X. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)

Hypotheses

Ref	Expression
dscmet.1	|- X e. _V
dscmet.2	|- D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = if(x = y, 0, 1))}

Assertion

Ref	Expression
dscmet	|- D e. Met

Distinct variable group:   x,X,y,z

Proof of Theorem dscmet

Step	Hyp	Ref	Expression
1		dscmet.1	. 2 |- X e. _V
2		dscmet.2	. . 3 |- D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = if(x = y, 0, 1))}
3		0re 6603	. . . . 5 |- 0 e. RR
4		1re 6598	. . . . 5 |- 1 e. RR
5	3, 4	keepel 3030	. . . 4 |- if(x = y, 0, 1) e. RR
6	5	a1i 8	. . 3 |- ((x e. X /\ y e. X) -> if(x = y, 0, 1) e. RR)
7	2, 6	foprab 5062	. 2 |- D:(X X. X)-->RR
8	3, 4	keepel 3030	. . . . . 6 |- if(w = v, 0, 1) e. RR
9	8	elisseti 2301	. . . . 5 |- if(w = v, 0, 1) e. _V
10		eqeq1 1890	. . . . . 6 |- (x = w -> (x = y <-> w = y))
11	10	ifbid 2996	. . . . 5 |- (x = w -> if(x = y, 0, 1) = if(w = y, 0, 1))
12		eqeq2 1893	. . . . . 6 |- (y = v -> (w = y <-> w = v))
13	12	ifbid 2996	. . . . 5 |- (y = v -> if(w = y, 0, 1) = if(w = v, 0, 1))
14	9, 11, 13, 2	oprabval2 4957	. . . 4 |- ((w e. X /\ v e. X) -> (wDv) = if(w = v, 0, 1))
15	14	eqeq1d 1892	. . 3 |- ((w e. X /\ v e. X) -> ((wDv) = 0 <-> if(w = v, 0, 1) = 0))
16		eqif 3004	. . . . . 6 |- (0 = if(w = v, 0, 1) <-> ((w = v /\ 0 = 0) \/ (-. w = v /\ 0 = 1)))
17		simpl 346	. . . . . . 7 |- ((w = v /\ 0 = 0) -> w = v)
18		ax1ne0 6433	. . . . . . . . . . . 12 |- 1 =/= 0
19		necom 2094	. . . . . . . . . . . 12 |- (1 =/= 0 <-> 0 =/= 1)
20	18, 19	mpbi 206	. . . . . . . . . . 11 |- 0 =/= 1
21		df-ne 2019	. . . . . . . . . . 11 |- (0 =/= 1 <-> -. 0 = 1)
22	20, 21	mpbi 206	. . . . . . . . . 10 |- -. 0 = 1
23	22	pm2.21i 93	. . . . . . . . 9 |- (0 = 1 -> (w = v \/ w = v))
24	23	orcanai 754	. . . . . . . 8 |- ((0 = 1 /\ -. w = v) -> w = v)
25	24	ancoms 484	. . . . . . 7 |- ((-. w = v /\ 0 = 1) -> w = v)
26	17, 25	jaoi 368	. . . . . 6 |- (((w = v /\ 0 = 0) \/ (-. w = v /\ 0 = 1)) -> w = v)
27	16, 26	sylbi 216	. . . . 5 |- (0 = if(w = v, 0, 1) -> w = v)
28	27	eqcoms 1887	. . . 4 |- (if(w = v, 0, 1) = 0 -> w = v)
29		iftrue 2989	. . . 4 |- (w = v -> if(w = v, 0, 1) = 0)
30	28, 29	impbii 174	. . 3 |- (if(w = v, 0, 1) = 0 <-> w = v)
31	15, 30	syl6bb 595	. 2 |- ((w e. X /\ v e. X) -> ((wDv) = 0 <-> w = v))
32		equtr 1490	. . . . . . . . 9 |- (u = w -> (w = v -> u = v))
33	32	imdistani 491	. . . . . . . 8 |- ((u = w /\ w = v) -> (u = w /\ u = v))
34		iftrue 2989	. . . . . . . . . 10 |- (u = w -> if(u = w, 0, 1) = 0)
35		iftrue 2989	. . . . . . . . . 10 |- (u = v -> if(u = v, 0, 1) = 0)
36	34, 35	opreqan12d 4902	. . . . . . . . 9 |- ((u = w /\ u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = (0 + 0))
37		0nn0 7322	. . . . . . . . . 10 |- 0 e. NN0
38	3, 37	nn0addge1i 7341	. . . . . . . . 9 |- 0 <_ (0 + 0)
39	36, 38	syl5breqr 3373	. . . . . . . 8 |- ((u = w /\ u = v) -> 0 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
40	33, 39	syl 12	. . . . . . 7 |- ((u = w /\ w = v) -> 0 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
41		equequ2 1495	. . . . . . . . . . 11 |- (w = v -> (u = w <-> u = v))
42	41	notbid 673	. . . . . . . . . 10 |- (w = v -> (-. u = w <-> -. u = v))
43	42	biimpcd 172	. . . . . . . . 9 |- (-. u = w -> (w = v -> -. u = v))
44	43	imdistani 491	. . . . . . . 8 |- ((-. u = w /\ w = v) -> (-. u = w /\ -. u = v))
45		iffalse 2991	. . . . . . . . . 10 |- (-. u = w -> if(u = w, 0, 1) = 1)
46		iffalse 2991	. . . . . . . . . 10 |- (-. u = v -> if(u = v, 0, 1) = 1)
47	45, 46	opreqan12d 4902	. . . . . . . . 9 |- ((-. u = w /\ -. u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = (1 + 1))
48		lt01 6871	. . . . . . . . . . 11 |- 0 < 1
49	3, 4, 48	ltleii 6756	. . . . . . . . . 10 |- 0 <_ 1
50	4, 4	addge0i 6777	. . . . . . . . . 10 |- ((0 <_ 1 /\ 0 <_ 1) -> 0 <_ (1 + 1))
51	49, 49, 50	mp2an 761	. . . . . . . . 9 |- 0 <_ (1 + 1)
52	47, 51	syl5breqr 3373	. . . . . . . 8 |- ((-. u = w /\ -. u = v) -> 0 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
53	44, 52	syl 12	. . . . . . 7 |- ((-. u = w /\ w = v) -> 0 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
54	40, 53	pm2.61ian 534	. . . . . 6 |- (w = v -> 0 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
55	29, 54	eqbrtrd 3357	. . . . 5 |- (w = v -> if(w = v, 0, 1) <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
56		iffalse 2991	. . . . . 6 |- (-. w = v -> if(w = v, 0, 1) = 1)
57		neeq1 2024	. . . . . . . . . . . 12 |- (u = w -> (u =/= v <-> w =/= v))
58	57	biimprd 171	. . . . . . . . . . 11 |- (u = w -> (w =/= v -> u =/= v))
59		df-ne 2019	. . . . . . . . . . 11 |- (w =/= v <-> -. w = v)
60		df-ne 2019	. . . . . . . . . . 11 |- (u =/= v <-> -. u = v)
61	58, 59, 60	3imtr3g 611	. . . . . . . . . 10 |- (u = w -> (-. w = v -> -. u = v))
62	61	imdistani 491	. . . . . . . . 9 |- ((u = w /\ -. w = v) -> (u = w /\ -. u = v))
63	34, 46	opreqan12d 4902	. . . . . . . . . . 11 |- ((u = w /\ -. u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = (0 + 1))
64		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
65	64	addid2i 6484	. . . . . . . . . . 11 |- (0 + 1) = 1
66	63, 65	syl6eq 1944	. . . . . . . . . 10 |- ((u = w /\ -. u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = 1)
67	4	leidi 6790	. . . . . . . . . 10 |- 1 <_ 1
68	66, 67	syl5breqr 3373	. . . . . . . . 9 |- ((u = w /\ -. u = v) -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
69	62, 68	syl 12	. . . . . . . 8 |- ((u = w /\ -. w = v) -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
70	69	ex 402	. . . . . . 7 |- (u = w -> (-. w = v -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1))))
71	45, 35	opreqan12d 4902	. . . . . . . . . . 11 |- ((-. u = w /\ u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = (1 + 0))
72	64	addid1i 6483	. . . . . . . . . . 11 |- (1 + 0) = 1
73	71, 72	syl6eq 1944	. . . . . . . . . 10 |- ((-. u = w /\ u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = 1)
74	73, 67	syl5breqr 3373	. . . . . . . . 9 |- ((-. u = w /\ u = v) -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
75		df-2 7154	. . . . . . . . . . 11 |- 2 = (1 + 1)
76	47, 75	syl6eqr 1946	. . . . . . . . . 10 |- ((-. u = w /\ -. u = v) -> (if(u = w, 0, 1) + if(u = v, 0, 1)) = 2)
77		2re 7163	. . . . . . . . . . 11 |- 2 e. RR
78		1lt2 7212	. . . . . . . . . . 11 |- 1 < 2
79	4, 77, 78	ltleii 6756	. . . . . . . . . 10 |- 1 <_ 2
80	76, 79	syl5breqr 3373	. . . . . . . . 9 |- ((-. u = w /\ -. u = v) -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
81	74, 80	pm2.61dan 535	. . . . . . . 8 |- (-. u = w -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
82	81	a1d 15	. . . . . . 7 |- (-. u = w -> (-. w = v -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1))))
83	70, 82	pm2.61i 140	. . . . . 6 |- (-. w = v -> 1 <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
84	56, 83	eqbrtrd 3357	. . . . 5 |- (-. w = v -> if(w = v, 0, 1) <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
85	55, 84	pm2.61i 140	. . . 4 |- if(w = v, 0, 1) <_ (if(u = w, 0, 1) + if(u = v, 0, 1))
86	85	a1i 8	. . 3 |- ((w e. X /\ v e. X /\ u e. X) -> if(w = v, 0, 1) <_ (if(u = w, 0, 1) + if(u = v, 0, 1)))
87	14	3adant3 896	. . 3 |- ((w e. X /\ v e. X /\ u e. X) -> (wDv) = if(w = v, 0, 1))
88	3, 4	keepel 3030	. . . . . . . 8 |- if(u = w, 0, 1) e. RR
89	88	elisseti 2301	. . . . . . 7 |- if(u = w, 0, 1) e. _V
90		eqeq1 1890	. . . . . . . 8 |- (x = u -> (x = y <-> u = y))
91	90	ifbid 2996	. . . . . . 7 |- (x = u -> if(x = y, 0, 1) = if(u = y, 0, 1))
92		eqeq2 1893	. . . . . . . 8 |- (y = w -> (u = y <-> u = w))
93	92	ifbid 2996	. . . . . . 7 |- (y = w -> if(u = y, 0, 1) = if(u = w, 0, 1))
94	89, 91, 93, 2	oprabval2 4957	. . . . . 6 |- ((u e. X /\ w e. X) -> (uDw) = if(u = w, 0, 1))
95	94	ancoms 484	. . . . 5 |- ((w e. X /\ u e. X) -> (uDw) = if(u = w, 0, 1))
96	95	3adant2 895	. . . 4 |- ((w e. X /\ v e. X /\ u e. X) -> (uDw) = if(u = w, 0, 1))
97	3, 4	keepel 3030	. . . . . . . 8 |- if(u = v, 0, 1) e. RR
98	97	elisseti 2301	. . . . . . 7 |- if(u = v, 0, 1) e. _V
99		eqeq2 1893	. . . . . . . 8 |- (y = v -> (u = y <-> u = v))
100	99	ifbid 2996	. . . . . . 7 |- (y = v -> if(u = y, 0, 1) = if(u = v, 0, 1))
101	98, 91, 100, 2	oprabval2 4957	. . . . . 6 |- ((u e. X /\ v e. X) -> (uDv) = if(u = v, 0, 1))
102	101	ancoms 484	. . . . 5 |- ((v e. X /\ u e. X) -> (uDv) = if(u = v, 0, 1))
103	102	3adant1 894	. . . 4 |- ((w e. X /\ v e. X /\ u e. X) -> (uDv) = if(u = v, 0, 1))
104	96, 103	opreq12d 4900	. . 3 |- ((w e. X /\ v e. X /\ u e. X) -> ((uDw) + (uDv)) = (if(u = w, 0, 1) + if(u = v, 0, 1)))
105	86, 87, 104	3brtr4d 3367	. 2 |- ((w e. X /\ v e. X /\ u e. X) -> (wDv) <_ ((uDw) + (uDv)))
106	1, 7, 31, 105	ismeti 9079	1 |- D e. Met

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  ifcif 2982   class class class wbr 3338  (class class class)co 4884  {copab2 4885  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   <_ cle 6448  2c2 7145  Metcme 9066

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-n0 7309  df-met 9070

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