Theorem metider 27624

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 27621	. . . 4 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
2		xpss 5109	. . . 4 |- ( X X. X ) C_ ( _V X. _V )
3	1, 2	syl6ss 3516	. . 3 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( _V X. _V ) )
4		df-rel 5006	. . 3 |- ( Rel (~Met ` D ) <-> (~Met ` D ) C_ ( _V X. _V ) )
5	3, 4	sylibr 212	. 2 |- ( D e. (PsMet ` X ) -> Rel (~Met ` D ) )
6	1	ssbrd 4488	. . . . 5 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) y -> x ( X X. X ) y ) )
7	6	imp 429	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> x ( X X. X ) y )
8		brxp 5030	. . . 4 |- ( x ( X X. X ) y <-> ( x e. X /\ y e. X ) )
9	7, 8	sylib 196	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) y ) -> ( x e. X /\ y e. X ) )
10		psmetsym 20641	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
11	10	3expb 1197	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x D y ) = ( y D x ) )
12	11	eqeq1d 2469	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) = 0 <-> ( y D x ) = 0 ) )
13		metidv 27622	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> ( x D y ) = 0 ) )
14		metidv 27622	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ x e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
15	14	ancom2s 800	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( y (~Met ` D ) x <-> ( y D x ) = 0 ) )
16	12, 13, 15	3bitr4d 285	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> y (~Met ` D ) x ) )
17	16	biimpd 207	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y -> y (~Met ` D ) x ) )
18	17	impancom 440	. . 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 457	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> D e. (PsMet ` X ) )
21		simprr 756	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y (~Met ` D ) z )
22	1	ssbrd 4488	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( y (~Met ` D ) z -> y ( X X. X ) z ) )
23	22	imp 429	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> y ( X X. X ) z )
24		brxp 5030	. . . . . . . . 9 |- ( y ( X X. X ) z <-> ( y e. X /\ z e. X ) )
25	23, 24	sylib 196	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ y (~Met ` D ) z ) -> ( y e. X /\ z e. X ) )
26	21, 25	syldan 470	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y e. X /\ z e. X ) )
27	26	simpld 459	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> y e. X )
28		simprl 755	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) y )
29	28, 9	syldan 470	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x e. X /\ y e. X ) )
30	29	simpld 459	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x e. X )
31	26	simprd 463	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> z e. X )
32		psmettri2 20640	. . . . . 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 1230	. . . . 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 470	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = ( y D x ) )
35	29, 13	syldan 470	. . . . . . . . 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 210	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D y ) = 0 )
37	34, 36	eqtr3d 2510	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D x ) = 0 )
38		metidv 27622	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( y e. X /\ z e. X ) ) -> ( y (~Met ` D ) z <-> ( y D z ) = 0 ) )
39	26, 38	syldan 470	. . . . . . . 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 210	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D z ) = 0 )
41	37, 40	oveq12d 6303	. . . . . 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 9641	. . . . . . 7 |- 0 e. RR*
43		xaddid1 11439	. . . . . . 7 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
44	42, 43	ax-mp 5	. . . . . 6 |- ( 0 +e 0 ) = 0
45	41, 44	syl6eq 2524	. . . . 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 4471	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) <_ 0 )
47		psmetge0 20643	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> 0 <_ ( x D z ) )
48	20, 30, 31, 47	syl3anc 1228	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> 0 <_ ( x D z ) )
49		psmetcl 20638	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> ( x D z ) e. RR* )
50	20, 30, 31, 49	syl3anc 1228	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) e. RR* )
51		xrletri3 11359	. . . . . 6 |- ( ( ( x D z ) e. RR* /\ 0 e. RR* ) -> ( ( x D z ) = 0 <-> ( ( x D z ) <_ 0 /\ 0 <_ ( x D z ) ) ) )
52	42, 51	mpan2 671	. . . . 5 |- ( ( x D z ) e. RR* -> ( ( x D z ) = 0 <-> ( ( x D z ) <_ 0 /\ 0 <_ ( x D z ) ) ) )
53	50, 52	syl 16	. . . 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 ) ) ) )
54	46, 48, 53	mpbir2and 920	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) = 0 )
55		metidv 27622	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ z e. X ) ) -> ( x (~Met ` D ) z <-> ( x D z ) = 0 ) )
56	20, 30, 31, 55	syl12anc 1226	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x (~Met ` D ) z <-> ( x D z ) = 0 ) )
57	54, 56	mpbird 232	. 2 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> x (~Met ` D ) z )
58		psmet0 20639	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x D x ) = 0 )
59		metidv 27622	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ x e. X ) ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
60	59	anabsan2 820	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
61	58, 60	mpbird 232	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x (~Met ` D ) x )
62	1	ssbrd 4488	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( x (~Met ` D ) x -> x ( X X. X ) x ) )
63	62	imp 429	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x ( X X. X ) x )
64		brxp 5030	. . . . 5 |- ( x ( X X. X ) x <-> ( x e. X /\ x e. X ) )
65	63, 64	sylib 196	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> ( x e. X /\ x e. X ) )
66	65	simpld 459	. . 3 |- ( ( D e. (PsMet ` X ) /\ x (~Met ` D ) x ) -> x e. X )
67	61, 66	impbida 830	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X <-> x (~Met ` D ) x ) )
68	5, 19, 57, 67	iserd 7338	1 |- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   C_ wss 3476   class class class wbr 4447   X. cxp 4997   Rel wrel 5004   ` cfv 5588  (class class class)co 6285   Er wer 7309   0cc0 9493   RR*cxr 9628   <_ cle 9630   +ecxad 11317  PsMetcpsmet 18213  ~Metcmetid 27616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-2 10595  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-psmet 18222  df-metid 27618

This theorem is referenced by:  pstmxmet  27627