Theorem metider 28690

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 28687	. . . 4 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
2		xpss 4940	. . . 4 |- ( X X. X ) C_ ( _V X. _V )
3	1, 2	syl6ss 3443	. . 3 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( _V X. _V ) )
4		df-rel 4840	. . 3 |- ( Rel (~Met ` D ) <-> (~Met ` D ) C_ ( _V X. _V ) )
5	3, 4	sylibr 216	. 2 |- ( D e. (PsMet ` X ) -> Rel (~Met ` D ) )
6	1	ssbrd 4443	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) y -> x ( X X. X ) y ) )
7	6	imp 431	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> x ( X X. X ) y )
8		brxp 4864	. . . 4 |- ( x ( X X. X ) y <-> ( x e. X /\ y e. X ) )
9	7, 8	sylib 200	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> ( x e. X /\ y e. X ) )
10		psmetsym 21319	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
11	10	3expb 1208	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x D y ) = ( y D x ) )
12	11	eqeq1d 2452	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) = 0 <-> ( y D x ) = 0 ) )
13		metidv 28688	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> ( x D y ) = 0 ) )
14		metidv 28688	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ x e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
15	14	ancom2s 810	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
16	12, 13, 15	3bitr4d 289	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> y (~Met ` D ) x ) )
17	16	biimpd 211	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y -> y (~Met ` D ) x ) )
18	17	impancom 442	. . 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 459	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> D e. (PsMet ` X ) )
21		simprr 765	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y (~Met ` D ) z )
22	1	ssbrd 4443	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( y (~Met ` D ) z -> y ( X X. X ) z ) )
23	22	imp 431	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> y ( X X. X ) z )
24		brxp 4864	. . . . . . . . 9 |- ( y ( X X. X ) z <-> ( y e. X /\ z e. X ) )
25	23, 24	sylib 200	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> ( y e. X /\ z e. X ) )
26	21, 25	syldan 473	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y e. X /\ z e. X ) )
27	26	simpld 461	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y e. X )
28		simprl 763	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) y )
29	28, 9	syldan 473	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x e. X /\ y e. X ) )
30	29	simpld 461	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x e. X )
31	26	simprd 465	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> z e. X )
32		psmettri2 21318	. . . . . 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 1269	. . . . 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 473	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = ( y D x ) )
35	29, 13	syldan 473	. . . . . . . . 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 214	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = 0 )
37	34, 36	eqtr3d 2486	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D x ) = 0 )
38		metidv 28688	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ z e. X ) ) -> ( y (~Met ` D ) z <-> ( y D z ) = 0 ) )
39	26, 38	syldan 473	. . . . . . . 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 214	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D z ) = 0 )
41	37, 40	oveq12d 6306	. . . . . 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 9684	. . . . . . 7 |- 0 e. RR*
43		xaddid1 11529	. . . . . . 7 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
44	42, 43	ax-mp 5	. . . . . 6 |- ( 0 +e 0 ) = 0
45	41, 44	syl6eq 2500	. . . . 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 4426	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) <_ 0 )
47		psmetge0 21321	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> 0 <_ ( x D z ) )
48	20, 30, 31, 47	syl3anc 1267	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> 0 <_ ( x D z ) )
49		psmetcl 21316	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> ( x D z ) e. RR* )
50	20, 30, 31, 49	syl3anc 1267	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) e. RR* )
51		xrletri3 11448	. . . . 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 667	. . . 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 932	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) = 0 )
54		metidv 28688	. . . 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 1265	. . 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 236	. 2 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) z )
57		psmet0 21317	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x D x ) = 0 )
58		metidv 28688	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ x e. X ) ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
59	58	anabsan2 830	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
60	57, 59	mpbird 236	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x (~Met ` D ) x )
61	1	ssbrd 4443	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) x -> x ( X X. X ) x ) )
62	61	imp 431	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x ( X X. X ) x )
63		brxp 4864	. . . . 5 |- ( x ( X X. X ) x <-> ( x e. X /\ x e. X ) )
64	62, 63	sylib 200	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> ( x e. X /\ x e. X ) )
65	64	simpld 461	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x e. X )
66	60, 65	impbida 842	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X <-> x (~Met ` D ) x ) )
67	5, 19, 56, 66	iserd 7386	1 |- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   _Vcvv 3044   C_ wss 3403   class class class wbr 4401   X. cxp 4831   Rel wrel 4838   ` cfv 5581  (class class class)co 6288   Er wer 7357   0cc0 9536   RR*cxr 9671   <_ cle 9673   +ecxad 11404  PsMetcpsmet 18947  ~Metcmetid 28682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-2 10665  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-psmet 18955  df-metid 28684

This theorem is referenced by:  pstmxmet  28693