Theorem dscmet 22377

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

Hypothesis

Ref	Expression
dscmet.1	|- D = ( x e. X , y e. X |-> if ( x = y , 0 , 1 ) )

Assertion

Ref	Expression
dscmet	|- ( X e. V -> D e. ( Met ` X ) )

Distinct variable group:   x, y, X

Allowed substitution hints:   D( x, y)   V( x, y)

Proof of Theorem dscmet

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 10040	. . . . . 6 |- 0 e. RR
2		1re 10039	. . . . . 6 |- 1 e. RR
3	1, 2	keepel 4155	. . . . 5 |- if ( x = y , 0 , 1 ) e. RR
4	3	rgen2w 2925	. . . 4 |- A. x e. X A. y e. X if ( x = y , 0 , 1 ) e. RR
5		dscmet.1	. . . . 5 |- D = ( x e. X , y e. X |-> if ( x = y , 0 , 1 ) )
6	5	fmpt2 7237	. . . 4 |- ( A. x e. X A. y e. X if ( x = y , 0 , 1 ) e. RR <-> D : ( X X. X ) --> RR )
7	4, 6	mpbi 220	. . 3 |- D : ( X X. X ) --> RR
8		equequ1 1952	. . . . . . . . 9 |- ( x = w -> ( x = y <-> w = y ) )
9	8	ifbid 4108	. . . . . . . 8 |- ( x = w -> if ( x = y , 0 , 1 ) = if ( w = y , 0 , 1 ) )
10		equequ2 1953	. . . . . . . . 9 |- ( y = v -> ( w = y <-> w = v ) )
11	10	ifbid 4108	. . . . . . . 8 |- ( y = v -> if ( w = y , 0 , 1 ) = if ( w = v , 0 , 1 ) )
12		0nn0 11307	. . . . . . . . . 10 |- 0 e. NN0
13		1nn0 11308	. . . . . . . . . 10 |- 1 e. NN0
14	12, 13	keepel 4155	. . . . . . . . 9 |- if ( w = v , 0 , 1 ) e. NN0
15	14	elexi 3213	. . . . . . . 8 |- if ( w = v , 0 , 1 ) e. _V
16	9, 11, 5, 15	ovmpt2 6796	. . . . . . 7 |- ( ( w e. X /\ v e. X ) -> ( w D v ) = if ( w = v , 0 , 1 ) )
17	16	eqeq1d 2624	. . . . . 6 |- ( ( w e. X /\ v e. X ) -> ( ( w D v ) = 0 <-> if ( w = v , 0 , 1 ) = 0 ) )
18		iffalse 4095	. . . . . . . . . 10 |- ( -. w = v -> if ( w = v , 0 , 1 ) = 1 )
19		ax-1ne0 10005	. . . . . . . . . . 11 |- 1 =/= 0
20	19	a1i 11	. . . . . . . . . 10 |- ( -. w = v -> 1 =/= 0 )
21	18, 20	eqnetrd 2861	. . . . . . . . 9 |- ( -. w = v -> if ( w = v , 0 , 1 ) =/= 0 )
22	21	neneqd 2799	. . . . . . . 8 |- ( -. w = v -> -. if ( w = v , 0 , 1 ) = 0 )
23	22	con4i 113	. . . . . . 7 |- ( if ( w = v , 0 , 1 ) = 0 -> w = v )
24		iftrue 4092	. . . . . . 7 |- ( w = v -> if ( w = v , 0 , 1 ) = 0 )
25	23, 24	impbii 199	. . . . . 6 |- ( if ( w = v , 0 , 1 ) = 0 <-> w = v )
26	17, 25	syl6bb 276	. . . . 5 |- ( ( w e. X /\ v e. X ) -> ( ( w D v ) = 0 <-> w = v ) )
27	12, 13	keepel 4155	. . . . . . . . . . 11 |- if ( u = w , 0 , 1 ) e. NN0
28	12, 13	keepel 4155	. . . . . . . . . . 11 |- if ( u = v , 0 , 1 ) e. NN0
29	27, 28	nn0addcli 11330	. . . . . . . . . 10 |- ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN0
30		elnn0 11294	. . . . . . . . . 10 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN0 <-> ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN \/ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 ) )
31	29, 30	mpbi 220	. . . . . . . . 9 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN \/ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 )
32		breq1 4656	. . . . . . . . . . . 12 |- ( 0 = if ( w = v , 0 , 1 ) -> ( 0 <_ 1 <-> if ( w = v , 0 , 1 ) <_ 1 ) )
33		breq1 4656	. . . . . . . . . . . 12 |- ( 1 = if ( w = v , 0 , 1 ) -> ( 1 <_ 1 <-> if ( w = v , 0 , 1 ) <_ 1 ) )
34		0le1 10551	. . . . . . . . . . . 12 |- 0 <_ 1
35	2	leidi 10562	. . . . . . . . . . . 12 |- 1 <_ 1
36	32, 33, 34, 35	keephyp 4152	. . . . . . . . . . 11 |- if ( w = v , 0 , 1 ) <_ 1
37		nnge1 11046	. . . . . . . . . . 11 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN -> 1 <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
38	14	nn0rei 11303	. . . . . . . . . . . 12 |- if ( w = v , 0 , 1 ) e. RR
39	29	nn0rei 11303	. . . . . . . . . . . 12 |- ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. RR
40	38, 2, 39	letri 10166	. . . . . . . . . . 11 |- ( ( if ( w = v , 0 , 1 ) <_ 1 /\ 1 <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) ) -> if ( w = v , 0 , 1 ) <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
41	36, 37, 40	sylancr 695	. . . . . . . . . 10 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN -> if ( w = v , 0 , 1 ) <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
42	27	nn0ge0i 11320	. . . . . . . . . . . . 13 |- 0 <_ if ( u = w , 0 , 1 )
43	28	nn0ge0i 11320	. . . . . . . . . . . . 13 |- 0 <_ if ( u = v , 0 , 1 )
44	27	nn0rei 11303	. . . . . . . . . . . . . 14 |- if ( u = w , 0 , 1 ) e. RR
45	28	nn0rei 11303	. . . . . . . . . . . . . 14 |- if ( u = v , 0 , 1 ) e. RR
46	44, 45	add20i 10571	. . . . . . . . . . . . 13 |- ( ( 0 <_ if ( u = w , 0 , 1 ) /\ 0 <_ if ( u = v , 0 , 1 ) ) -> ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 <-> ( if ( u = w , 0 , 1 ) = 0 /\ if ( u = v , 0 , 1 ) = 0 ) ) )
47	42, 43, 46	mp2an 708	. . . . . . . . . . . 12 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 <-> ( if ( u = w , 0 , 1 ) = 0 /\ if ( u = v , 0 , 1 ) = 0 ) )
48		equequ2 1953	. . . . . . . . . . . . . . . . . . 19 |- ( v = w -> ( u = v <-> u = w ) )
49	48	ifbid 4108	. . . . . . . . . . . . . . . . . 18 |- ( v = w -> if ( u = v , 0 , 1 ) = if ( u = w , 0 , 1 ) )
50	49	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( v = w -> ( if ( u = v , 0 , 1 ) = 0 <-> if ( u = w , 0 , 1 ) = 0 ) )
51	50, 48	bibi12d 335	. . . . . . . . . . . . . . . 16 |- ( v = w -> ( ( if ( u = v , 0 , 1 ) = 0 <-> u = v ) <-> ( if ( u = w , 0 , 1 ) = 0 <-> u = w ) ) )
52		equequ1 1952	. . . . . . . . . . . . . . . . . . . 20 |- ( w = u -> ( w = v <-> u = v ) )
53	52	ifbid 4108	. . . . . . . . . . . . . . . . . . 19 |- ( w = u -> if ( w = v , 0 , 1 ) = if ( u = v , 0 , 1 ) )
54	53	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( w = u -> ( if ( w = v , 0 , 1 ) = 0 <-> if ( u = v , 0 , 1 ) = 0 ) )
55	54, 52	bibi12d 335	. . . . . . . . . . . . . . . . 17 |- ( w = u -> ( ( if ( w = v , 0 , 1 ) = 0 <-> w = v ) <-> ( if ( u = v , 0 , 1 ) = 0 <-> u = v ) ) )
56	55, 25	chvarv 2263	. . . . . . . . . . . . . . . 16 |- ( if ( u = v , 0 , 1 ) = 0 <-> u = v )
57	51, 56	chvarv 2263	. . . . . . . . . . . . . . 15 |- ( if ( u = w , 0 , 1 ) = 0 <-> u = w )
58		eqtr2 2642	. . . . . . . . . . . . . . 15 |- ( ( u = w /\ u = v ) -> w = v )
59	57, 56, 58	syl2anb 496	. . . . . . . . . . . . . 14 |- ( ( if ( u = w , 0 , 1 ) = 0 /\ if ( u = v , 0 , 1 ) = 0 ) -> w = v )
60	59	iftrued 4094	. . . . . . . . . . . . 13 |- ( ( if ( u = w , 0 , 1 ) = 0 /\ if ( u = v , 0 , 1 ) = 0 ) -> if ( w = v , 0 , 1 ) = 0 )
61	1	leidi 10562	. . . . . . . . . . . . 13 |- 0 <_ 0
62	60, 61	syl6eqbr 4692	. . . . . . . . . . . 12 |- ( ( if ( u = w , 0 , 1 ) = 0 /\ if ( u = v , 0 , 1 ) = 0 ) -> if ( w = v , 0 , 1 ) <_ 0 )
63	47, 62	sylbi 207	. . . . . . . . . . 11 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 -> if ( w = v , 0 , 1 ) <_ 0 )
64		id 22	. . . . . . . . . . 11 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 -> ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 )
65	63, 64	breqtrrd 4681	. . . . . . . . . 10 |- ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 -> if ( w = v , 0 , 1 ) <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
66	41, 65	jaoi 394	. . . . . . . . 9 |- ( ( ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) e. NN \/ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) = 0 ) -> if ( w = v , 0 , 1 ) <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
67	31, 66	mp1i 13	. . . . . . . 8 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> if ( w = v , 0 , 1 ) <_ ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
68	16	adantl 482	. . . . . . . 8 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> ( w D v ) = if ( w = v , 0 , 1 ) )
69		eqeq12 2635	. . . . . . . . . . . 12 |- ( ( x = u /\ y = w ) -> ( x = y <-> u = w ) )
70	69	ifbid 4108	. . . . . . . . . . 11 |- ( ( x = u /\ y = w ) -> if ( x = y , 0 , 1 ) = if ( u = w , 0 , 1 ) )
71	27	elexi 3213	. . . . . . . . . . 11 |- if ( u = w , 0 , 1 ) e. _V
72	70, 5, 71	ovmpt2a 6791	. . . . . . . . . 10 |- ( ( u e. X /\ w e. X ) -> ( u D w ) = if ( u = w , 0 , 1 ) )
73	72	adantrr 753	. . . . . . . . 9 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> ( u D w ) = if ( u = w , 0 , 1 ) )
74		eqeq12 2635	. . . . . . . . . . . 12 |- ( ( x = u /\ y = v ) -> ( x = y <-> u = v ) )
75	74	ifbid 4108	. . . . . . . . . . 11 |- ( ( x = u /\ y = v ) -> if ( x = y , 0 , 1 ) = if ( u = v , 0 , 1 ) )
76	28	elexi 3213	. . . . . . . . . . 11 |- if ( u = v , 0 , 1 ) e. _V
77	75, 5, 76	ovmpt2a 6791	. . . . . . . . . 10 |- ( ( u e. X /\ v e. X ) -> ( u D v ) = if ( u = v , 0 , 1 ) )
78	77	adantrl 752	. . . . . . . . 9 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> ( u D v ) = if ( u = v , 0 , 1 ) )
79	73, 78	oveq12d 6668	. . . . . . . 8 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> ( ( u D w ) + ( u D v ) ) = ( if ( u = w , 0 , 1 ) + if ( u = v , 0 , 1 ) ) )
80	67, 68, 79	3brtr4d 4685	. . . . . . 7 |- ( ( u e. X /\ ( w e. X /\ v e. X ) ) -> ( w D v ) <_ ( ( u D w ) + ( u D v ) ) )
81	80	expcom 451	. . . . . 6 |- ( ( w e. X /\ v e. X ) -> ( u e. X -> ( w D v ) <_ ( ( u D w ) + ( u D v ) ) ) )
82	81	ralrimiv 2965	. . . . 5 |- ( ( w e. X /\ v e. X ) -> A. u e. X ( w D v ) <_ ( ( u D w ) + ( u D v ) ) )
83	26, 82	jca 554	. . . 4 |- ( ( w e. X /\ v e. X ) -> ( ( ( w D v ) = 0 <-> w = v ) /\ A. u e. X ( w D v ) <_ ( ( u D w ) + ( u D v ) ) ) )
84	83	rgen2a 2977	. . 3 |- A. w e. X A. v e. X ( ( ( w D v ) = 0 <-> w = v ) /\ A. u e. X ( w D v ) <_ ( ( u D w ) + ( u D v ) ) )
85	7, 84	pm3.2i 471	. 2 |- ( D : ( X X. X ) --> RR /\ A. w e. X A. v e. X ( ( ( w D v ) = 0 <-> w = v ) /\ A. u e. X ( w D v ) <_ ( ( u D w ) + ( u D v ) ) ) )
86		ismet 22128	. 2 |- ( X e. V -> ( D e. ( Met ` X ) <-> ( D : ( X X. X ) --> RR /\ A. w e. X A. v e. X ( ( ( w D v ) = 0 <-> w = v ) /\ A. u e. X ( w D v ) <_ ( ( u D w ) + ( u D v ) ) ) ) ) )
87	85, 86	mpbiri 248	1 |- ( X e. V -> D e. ( Met ` X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   NNcn 11020   NN0cn0 11292   Metcme 19732

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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-met 19740

This theorem is referenced by:  dscopn  22378