Theorem pstmval 29938

Description: Value of the metric induced by a pseudometric D. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Hypothesis

Ref	Expression
pstmval.1	|- .~ = (~Met ` D )

Assertion

Ref	Expression
pstmval	|- ( D e. (PsMet ` X ) -> (pstoMet ` D ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) )

Distinct variable groups:   a, b, x, y, z, D   X, a, b, x, y, z   .~ , a, b, x, y, z

Proof of Theorem pstmval

Dummy variables c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pstm 29932	. . 3 |- pstoMet = ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )
2	1	a1i 11	. 2 |- ( D e. (PsMet ` X ) -> pstoMet = ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) ) )
3		psmetdmdm 22110	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X = dom dom D )
4	3	adantr 481	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> X = dom dom D )
5		dmeq 5324	. . . . . . . . 9 |- ( d = D -> dom d = dom D )
6	5	dmeqd 5326	. . . . . . . 8 |- ( d = D -> dom dom d = dom dom D )
7	6	adantl 482	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = dom dom D )
8	4, 7	eqtr4d 2659	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> X = dom dom d )
9		qseq1 7796	. . . . . 6 |- ( X = dom dom d -> ( X /. .~ ) = ( dom dom d /. .~ ) )
10	8, 9	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( X /. .~ ) = ( dom dom d /. .~ ) )
11		fveq2 6191	. . . . . . . 8 |- ( d = D -> (~Met ` d ) = (~Met ` D ) )
12		pstmval.1	. . . . . . . 8 |- .~ = (~Met ` D )
13	11, 12	syl6reqr 2675	. . . . . . 7 |- ( d = D -> .~ = (~Met ` d ) )
14		qseq2 7797	. . . . . . 7 |- ( .~ = (~Met ` d ) -> ( dom dom d /. .~ ) = ( dom dom d /. (~Met ` d ) ) )
15	13, 14	syl 17	. . . . . 6 |- ( d = D -> ( dom dom d /. .~ ) = ( dom dom d /. (~Met ` d ) ) )
16	15	adantl 482	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( dom dom d /. .~ ) = ( dom dom d /. (~Met ` d ) ) )
17	10, 16	eqtr2d 2657	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( dom dom d /. (~Met ` d ) ) = ( X /. .~ ) )
18		mpt2eq12 6715	. . . 4 |- ( ( ( dom dom d /. (~Met ` d ) ) = ( X /. .~ ) /\ ( dom dom d /. (~Met ` d ) ) = ( X /. .~ ) ) -> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )
19	17, 17, 18	syl2anc 693	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )
20		simp1r 1086	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> d = D )
21	20	oveqd 6667	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> ( x d y ) = ( x D y ) )
22	21	eqeq2d 2632	. . . . . . 7 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> ( z = ( x d y ) <-> z = ( x D y ) ) )
23	22	2rexbidv 3057	. . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> ( E. x e. a E. y e. b z = ( x d y ) <-> E. x e. a E. y e. b z = ( x D y ) ) )
24	23	abbidv 2741	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> { z | E. x e. a E. y e. b z = ( x d y ) } = { z | E. x e. a E. y e. b z = ( x D y ) } )
25	24	unieqd 4446	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ d = D ) /\ a e. ( X /. .~ ) /\ b e. ( X /. .~ ) ) -> U. { z | E. x e. a E. y e. b z = ( x d y ) } = U. { z | E. x e. a E. y e. b z = ( x D y ) } )
26	25	mpt2eq3dva 6719	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) )
27	19, 26	eqtrd 2656	. 2 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) )
28		elfvdm 6220	. . . 4 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
29		fveq2 6191	. . . . . 6 |- ( x = X -> (PsMet ` x ) = (PsMet ` X ) )
30	29	eleq2d 2687	. . . . 5 |- ( x = X -> ( D e. (PsMet ` x ) <-> D e. (PsMet ` X ) ) )
31	30	rspcev 3309	. . . 4 |- ( ( X e. dom PsMet /\ D e. (PsMet ` X ) ) -> E. x e. dom PsMet D e. (PsMet ` x ) )
32	28, 31	mpancom 703	. . 3 |- ( D e. (PsMet ` X ) -> E. x e. dom PsMet D e. (PsMet ` x ) )
33		df-psmet 19738	. . . . 5 |- PsMet = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. a e. x ( ( a d a ) = 0 /\ A. b e. x A. c e. x ( a d b ) <_ ( ( c d a ) +e ( c d b ) ) ) } )
34	33	funmpt2 5927	. . . 4 |- Fun PsMet
35		elunirn 6509	. . . 4 |- ( Fun PsMet -> ( D e. U. ran PsMet <-> E. x e. dom PsMet D e. (PsMet ` x ) ) )
36	34, 35	ax-mp 5	. . 3 |- ( D e. U. ran PsMet <-> E. x e. dom PsMet D e. (PsMet ` x ) )
37	32, 36	sylibr 224	. 2 |- ( D e. (PsMet ` X ) -> D e. U. ran PsMet )
38		elfvex 6221	. . . 4 |- ( D e. (PsMet ` X ) -> X e. _V )
39		qsexg 7805	. . . 4 |- ( X e. _V -> ( X /. .~ ) e. _V )
40	38, 39	syl 17	. . 3 |- ( D e. (PsMet ` X ) -> ( X /. .~ ) e. _V )
41		mpt2exga 7246	. . 3 |- ( ( ( X /. .~ ) e. _V /\ ( X /. .~ ) e. _V ) -> ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) e. _V )
42	40, 40, 41	syl2anc 693	. 2 |- ( D e. (PsMet ` X ) -> ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) e. _V )
43	2, 27, 37, 42	fvmptd 6288	1 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) = ( a e. ( X /. .~ ) , b e. ( X /. .~ ) |-> U. { z | E. x e. a E. y e. b z = ( x D y ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   /.cqs 7741   ^m cmap 7857   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730  ~Metcmetid 29929  pstoMetcpstm 29930

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ec 7744  df-qs 7748  df-map 7859  df-xr 10078  df-psmet 19738  df-pstm 29932

This theorem is referenced by:  pstmfval  29939  pstmxmet  29940