Theorem metider 26343

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 26340	. . . 4 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
2		xpss 4967	. . . 4 |- ( X X. X ) C_ ( _V X. _V )
3	1, 2	syl6ss 3389	. . 3 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( _V X. _V ) )
4		df-rel 4868	. . 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 4354	. . . . 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 4891	. . . 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 19908	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ y e. X ) -> ( x D y ) = ( y D x ) )
11	10	3expb 1188	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x D y ) = ( y D x ) )
12	11	eqeq1d 2451	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) = 0 <-> ( y D x ) = 0 ) )
13		metidv 26341	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ y e. X ) ) -> ( x (~Met ` D ) y <-> ( x D y ) = 0 ) )
14		metidv 26341	. . . . . . 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 4354	. . . . . . . . . 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 4891	. . . . . . . . 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 19907	. . . . . 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 1220	. . . . 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 2477	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( y D x ) = 0 )
38		metidv 26341	. . . . . . . . 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 6130	. . . . . 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 9451	. . . . . . 7 |- 0 e. RR*
43		xaddid1 11230	. . . . . . 7 |- ( 0 e. RR* -> ( 0 +e 0 ) = 0 )
44	42, 43	ax-mp 5	. . . . . 6 |- ( 0 +e 0 ) = 0
45	41, 44	syl6eq 2491	. . . . 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 4337	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) <_ 0 )
47		psmetge0 19910	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> 0 <_ ( x D z ) )
48	20, 30, 31, 47	syl3anc 1218	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> 0 <_ ( x D z ) )
49		psmetcl 19905	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ z e. X ) -> ( x D z ) e. RR* )
50	20, 30, 31, 49	syl3anc 1218	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) e. RR* )
51		xrletri3 11150	. . . . . 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 913	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x (~Met ` D ) y /\ y (~Met ` D ) z ) ) -> ( x D z ) = 0 )
55		metidv 26341	. . . 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 1216	. . 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 19906	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x D x ) = 0 )
59		metidv 26341	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ x e. X ) ) -> ( x (~Met ` D ) x <-> ( x D x ) = 0 ) )
60	59	anabsan2 818	. . . 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 4354	. . . . . 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 4891	. . . . 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 828	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X <-> x (~Met ` D ) x ) )
68	5, 19, 57, 67	iserd 7148	1 |- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2993   C_ wss 3349   class class class wbr 4313   X. cxp 4859   Rel wrel 4866   ` cfv 5439  (class class class)co 6112   Er wer 7119   0cc0 9303   RR*cxr 9438   <_ cle 9440   +ecxad 11108  PsMetcpsmet 17822  ~Metcmetid 26335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-2 10401  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-psmet 17831  df-metid 26337

This theorem is referenced by:  pstmxmet  26346