Theorem metider 29937

Description: The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
metider	|- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )

Proof of Theorem metider

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metidss 29934	. . . 4 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
2		xpss 5226	. . . 4 |- ( X X. X ) C_ ( _V X. _V )
3	1, 2	syl6ss 3615	. . 3 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( _V X. _V ) )
4		df-rel 5121	. . 3 |- ( Rel (~Met ` D ) <-> (~Met ` D ) C_ ( _V X. _V ) )
5	3, 4	sylibr 224	. 2 |- ( D e. (PsMet ` X ) -> Rel (~Met ` D ) )
6	1	ssbrd 4696	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) y -> x ( X X. X ) y ) )
7	6	imp 445	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> x ( X X. X ) y )
8		brxp 5147	. . . 4 |- ( x ( X X. X ) y <-> ( x e. X /\ y e. X ) )
9	7, 8	sylib 208	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> ( x e. X /\ y e. X ) )
10		psmetsym 22115	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
11	10	3expb 1266	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x D y ) = ( y D x ) )
12	11	eqeq1d 2624	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) = 0 <-> ( y D x ) = 0 ) )
13		metidv 29935	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> ( x D y ) = 0 ) )
14		metidv 29935	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ x e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
15	14	ancom2s 844	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
16	12, 13, 15	3bitr4d 300	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> y (~Met ` D ) x ) )
17	16	biimpd 219	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y -> y (~Met ` D ) x ) )
18	17	impancom 456	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> ( ( x e. X /\ y e. X ) -> y (~Met ` D ) x ) )
19	9, 18	mpd 15	. 2 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> y (~Met ` D ) x )
20		simpl 473	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> D e. (PsMet ` X ) )
21		simprr 796	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y (~Met ` D ) z )
22	1	ssbrd 4696	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( y (~Met ` D ) z -> y ( X X. X ) z ) )
23	22	imp 445	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> y ( X X. X ) z )
24		brxp 5147	. . . . . . . . 9 |- ( y ( X X. X ) z <-> ( y e. X /\ z e. X ) )
25	23, 24	sylib 208	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> ( y e. X /\ z e. X ) )
26	21, 25	syldan 487	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y e. X /\ z e. X ) )
27	26	simpld 475	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y e. X )
28		simprl 794	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) y )
29	28, 9	syldan 487	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x e. X /\ y e. X ) )
30	29	simpld 475	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x e. X )
31	26	simprd 479	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> z e. X )
32		psmettri2 22114	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ x e. X /\ z e. X ) ) -> ( x D z ) <_ ( ( y D x ) +e ( y D z ) ) )
33	20, 27, 30, 31, 32	syl13anc 1328	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) <_ ( ( y D x ) +e ( y D z ) ) )
34	29, 11	syldan 487	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = ( y D x ) )
35	29, 13	syldan 487	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x (~Met ` D ) y <-> ( x D y ) = 0 ) )
36	28, 35	mpbid 222	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = 0 )
37	34, 36	eqtr3d 2658	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D x ) = 0 )
38		metidv 29935	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ z e. X ) ) -> ( y (~Met ` D ) z <-> ( y D z ) = 0 ) )
39	26, 38	syldan 487	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y (~Met ` D ) z <-> ( y D z ) = 0 ) )
40	21, 39	mpbid 222	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D z ) = 0 )
41	37, 40	oveq12d 6668	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( ( y D x ) +e ( y D z ) ) = ( 0 +e 0 ) )
42		0xr 10086	. . . . . . 7 |- 0 e. RR*
43		xaddid1 12072	. . . . . . 7 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
44	42, 43	ax-mp 5	. . . . . 6 |- ( 0 +e 0 ) = 0
45	41, 44	syl6eq 2672	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( ( y D x ) +e ( y D z ) ) = 0 )
46	33, 45	breqtrd 4679	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) <_ 0 )
47		psmetge0 22117	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> 0 <_ ( x D z ) )
48	20, 30, 31, 47	syl3anc 1326	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> 0 <_ ( x D z ) )
49		psmetcl 22112	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> ( x D z ) e. RR* )
50	20, 30, 31, 49	syl3anc 1326	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) e. RR* )
51		xrletri3 11985	. . . . 5 |- ( ( ( x D z ) e. RR* /\ 0 e. RR* ) -> ( ( x D z ) = 0 <-> ( ( x D z ) <_ 0 /\ 0 <_ ( x D z ) ) ) )
52	50, 42, 51	sylancl 694	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( ( x D z ) = 0 <-> ( ( x D z ) <_ 0 /\ 0 <_ ( x D z ) ) ) )
53	46, 48, 52	mpbir2and 957	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) = 0 )
54		metidv 29935	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ z e. X ) ) -> ( x (~Met ` D ) z <-> ( x D z ) = 0 ) )
55	20, 30, 31, 54	syl12anc 1324	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x (~Met ` D ) z <-> ( x D z ) = 0 ) )
56	53, 55	mpbird 247	. 2 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) z )
57		psmet0 22113	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x D x ) = 0 )
58		metidv 29935	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ x e. X ) ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
59	58	anabsan2 863	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
60	57, 59	mpbird 247	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x (~Met ` D ) x )
61	1	ssbrd 4696	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) x -> x ( X X. X ) x ) )
62	61	imp 445	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x ( X X. X ) x )
63		brxp 5147	. . . . 5 |- ( x ( X X. X ) x <-> ( x e. X /\ x e. X ) )
64	62, 63	sylib 208	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> ( x e. X /\ x e. X ) )
65	64	simpld 475	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x e. X )
66	60, 65	impbida 877	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X <-> x (~Met ` D ) x ) )
67	5, 19, 56, 66	iserd 7768	1 |- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   Er wer 7739   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730  ~Metcmetid 29929

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-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-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-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-psmet 19738  df-metid 29931

This theorem is referenced by:  pstmxmet  29940