Theorem pstmxmet 29940

Description: The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

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

Assertion

Ref	Expression
pstmxmet	|- ( D e. (PsMet ` X ) -> (pstoMet ` D ) e. ( *Met ` ( X /. .~ ) ) )

Proof of Theorem pstmxmet

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) = ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } )
2		vex 3203	. . . . . . 7 |- x e. _V
3		vex 3203	. . . . . . 7 |- y e. _V
4	2, 3	ab2rexex 7159	. . . . . 6 |- { z | E. a e. x E. b e. y z = ( a D b ) } e. _V
5	4	uniex 6953	. . . . 5 |- U. { z | E. a e. x E. b e. y z = ( a D b ) } e. _V
6	1, 5	fnmpt2i 7239	. . . 4 |- ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) Fn ( ( X /. .~ ) X. ( X /. .~ ) )
7		pstmval.1	. . . . . 6 |- .~ = (~Met ` D )
8	7	pstmval 29938	. . . . 5 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) = ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) )
9	8	fneq1d 5981	. . . 4 |- ( D e. (PsMet ` X ) -> ( (pstoMet ` D ) Fn ( ( X /. .~ ) X. ( X /. .~ ) ) <-> ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) Fn ( ( X /. .~ ) X. ( X /. .~ ) ) ) )
10	6, 9	mpbiri 248	. . 3 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) Fn ( ( X /. .~ ) X. ( X /. .~ ) ) )
11		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> x = [ a ] .~ )
12		simpr 477	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> y = [ b ] .~ )
13	11, 12	oveq12d 6668	. . . . . . . 8 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x (pstoMet ` D ) y ) = ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) )
14		simp-5l 808	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> D e. (PsMet ` X ) )
15		simp-4r 807	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> a e. X )
16		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> b e. X )
17	7	pstmfval 29939	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ a e. X /\ b e. X ) -> ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = ( a D b ) )
18	14, 15, 16, 17	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = ( a D b ) )
19	13, 18	eqtrd 2656	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x (pstoMet ` D ) y ) = ( a D b ) )
20		psmetf 22111	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
21	14, 20	syl 17	. . . . . . . 8 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> D : ( X X. X ) --> RR* )
22	21, 15, 16	fovrnd 6806	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( a D b ) e. RR* )
23	19, 22	eqeltrd 2701	. . . . . 6 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x (pstoMet ` D ) y ) e. RR* )
24		elqsi 7800	. . . . . . . 8 |- ( y e. ( X /. .~ ) -> E. b e. X y = [ b ] .~ )
25	24	ad2antll 765	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> E. b e. X y = [ b ] .~ )
26	25	ad2antrr 762	. . . . . 6 |- ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> E. b e. X y = [ b ] .~ )
27	23, 26	r19.29a 3078	. . . . 5 |- ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> ( x (pstoMet ` D ) y ) e. RR* )
28		elqsi 7800	. . . . . 6 |- ( x e. ( X /. .~ ) -> E. a e. X x = [ a ] .~ )
29	28	ad2antrl 764	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> E. a e. X x = [ a ] .~ )
30	27, 29	r19.29a 3078	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> ( x (pstoMet ` D ) y ) e. RR* )
31	30	ralrimivva 2971	. . 3 |- ( D e. (PsMet ` X ) -> A. x e. ( X /. .~ ) A. y e. ( X /. .~ ) ( x (pstoMet ` D ) y ) e. RR* )
32		ffnov 6764	. . 3 |- ( (pstoMet ` D ) : ( ( X /. .~ ) X. ( X /. .~ ) ) --> RR* <-> ( (pstoMet ` D ) Fn ( ( X /. .~ ) X. ( X /. .~ ) ) /\ A. x e. ( X /. .~ ) A. y e. ( X /. .~ ) ( x (pstoMet ` D ) y ) e. RR* ) )
33	10, 31, 32	sylanbrc 698	. 2 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) : ( ( X /. .~ ) X. ( X /. .~ ) ) --> RR* )
34	17	3expa 1265	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = ( a D b ) )
35	34	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 <-> ( a D b ) = 0 ) )
36	7	breqi 4659	. . . . . . . . . . . 12 |- ( a .~ b <-> a (~Met ` D ) b )
37		metidv 29935	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ ( a e. X /\ b e. X ) ) -> ( a (~Met ` D ) b <-> ( a D b ) = 0 ) )
38	37	anassrs 680	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( a (~Met ` D ) b <-> ( a D b ) = 0 ) )
39	36, 38	syl5bb 272	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( a .~ b <-> ( a D b ) = 0 ) )
40		metider 29937	. . . . . . . . . . . . . 14 |- ( D e. (PsMet ` X ) -> (~Met ` D ) Er X )
41	40	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> (~Met ` D ) Er X )
42		ereq1 7749	. . . . . . . . . . . . . 14 |- ( .~ = (~Met ` D ) -> ( .~ Er X <-> (~Met ` D ) Er X ) )
43	7, 42	ax-mp 5	. . . . . . . . . . . . 13 |- ( .~ Er X <-> (~Met ` D ) Er X )
44	41, 43	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> .~ Er X )
45		simplr 792	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> a e. X )
46	44, 45	erth 7791	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( a .~ b <-> [ a ] .~ = [ b ] .~ ) )
47	35, 39, 46	3bitr2d 296	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ b e. X ) -> ( ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 <-> [ a ] .~ = [ b ] .~ ) )
48	47	adantllr 755	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ b e. X ) -> ( ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 <-> [ a ] .~ = [ b ] .~ ) )
49	48	adantlr 751	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) -> ( ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 <-> [ a ] .~ = [ b ] .~ ) )
50	49	adantr 481	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 <-> [ a ] .~ = [ b ] .~ ) )
51	13	eqeq1d 2624	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( ( x (pstoMet ` D ) y ) = 0 <-> ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = 0 ) )
52	11, 12	eqeq12d 2637	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x = y <-> [ a ] .~ = [ b ] .~ ) )
53	50, 51, 52	3bitr4d 300	. . . . . 6 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) )
54	53, 26	r19.29a 3078	. . . . 5 |- ( ( ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) )
55	54, 29	r19.29a 3078	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) )
56		simp-6l 810	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> D e. (PsMet ` X ) )
57		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> c e. X )
58		simp-6r 811	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> a e. X )
59		simp-4r 807	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> b e. X )
60		psmettri2 22114	. . . . . . . . . . . . . 14 |- ( ( D e. (PsMet ` X ) /\ ( c e. X /\ a e. X /\ b e. X ) ) -> ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) )
61	56, 57, 58, 59, 60	syl13anc 1328	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) )
62		simp-5r 809	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> x = [ a ] .~ )
63		simpllr 799	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> y = [ b ] .~ )
64	62, 63	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( x (pstoMet ` D ) y ) = ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) )
65	56, 58, 59, 17	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( [ a ] .~ (pstoMet ` D ) [ b ] .~ ) = ( a D b ) )
66	64, 65	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( x (pstoMet ` D ) y ) = ( a D b ) )
67		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> z = [ c ] .~ )
68	67, 62	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( z (pstoMet ` D ) x ) = ( [ c ] .~ (pstoMet ` D ) [ a ] .~ ) )
69	7	pstmfval 29939	. . . . . . . . . . . . . . . 16 |- ( ( D e. (PsMet ` X ) /\ c e. X /\ a e. X ) -> ( [ c ] .~ (pstoMet ` D ) [ a ] .~ ) = ( c D a ) )
70	56, 57, 58, 69	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( [ c ] .~ (pstoMet ` D ) [ a ] .~ ) = ( c D a ) )
71	68, 70	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( z (pstoMet ` D ) x ) = ( c D a ) )
72	67, 63	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( z (pstoMet ` D ) y ) = ( [ c ] .~ (pstoMet ` D ) [ b ] .~ ) )
73	7	pstmfval 29939	. . . . . . . . . . . . . . . 16 |- ( ( D e. (PsMet ` X ) /\ c e. X /\ b e. X ) -> ( [ c ] .~ (pstoMet ` D ) [ b ] .~ ) = ( c D b ) )
74	56, 57, 59, 73	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( [ c ] .~ (pstoMet ` D ) [ b ] .~ ) = ( c D b ) )
75	72, 74	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( z (pstoMet ` D ) y ) = ( c D b ) )
76	71, 75	oveq12d 6668	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) = ( ( c D a ) +e ( c D b ) ) )
77	61, 66, 76	3brtr4d 4685	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
78	77	adantl6r 789	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) /\ c e. X ) /\ z = [ c ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
79		elqsi 7800	. . . . . . . . . . . 12 |- ( z e. ( X /. .~ ) -> E. c e. X z = [ c ] .~ )
80	79	ad5antlr 771	. . . . . . . . . . 11 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> E. c e. X z = [ c ] .~ )
81	78, 80	r19.29a 3078	. . . . . . . . . 10 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
82	81	adantl5r 788	. . . . . . . . 9 |- ( ( ( ( ( ( ( D e. (PsMet ` X ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) /\ b e. X ) /\ y = [ b ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
83	24	ad4antlr 769	. . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> E. b e. X y = [ b ] .~ )
84	82, 83	r19.29a 3078	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
85	84	adantl4r 787	. . . . . . 7 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ x e. ( X /. .~ ) ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) /\ a e. X ) /\ x = [ a ] .~ ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
86	28	ad3antlr 767	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ x e. ( X /. .~ ) ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) -> E. a e. X x = [ a ] .~ )
87	85, 86	r19.29a 3078	. . . . . 6 |- ( ( ( ( D e. (PsMet ` X ) /\ x e. ( X /. .~ ) ) /\ y e. ( X /. .~ ) ) /\ z e. ( X /. .~ ) ) -> ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
88	87	ralrimiva 2966	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ x e. ( X /. .~ ) ) /\ y e. ( X /. .~ ) ) -> A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
89	88	anasss 679	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) )
90	55, 89	jca 554	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( x e. ( X /. .~ ) /\ y e. ( X /. .~ ) ) ) -> ( ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) /\ A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) ) )
91	90	ralrimivva 2971	. 2 |- ( D e. (PsMet ` X ) -> A. x e. ( X /. .~ ) A. y e. ( X /. .~ ) ( ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) /\ A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) ) )
92		elfvex 6221	. . 3 |- ( D e. (PsMet ` X ) -> X e. _V )
93		qsexg 7805	. . 3 |- ( X e. _V -> ( X /. .~ ) e. _V )
94		isxmet 22129	. . 3 |- ( ( X /. .~ ) e. _V -> ( (pstoMet ` D ) e. ( *Met ` ( X /. .~ ) ) <-> ( (pstoMet ` D ) : ( ( X /. .~ ) X. ( X /. .~ ) ) --> RR* /\ A. x e. ( X /. .~ ) A. y e. ( X /. .~ ) ( ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) /\ A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) ) ) ) )
95	92, 93, 94	3syl 18	. 2 |- ( D e. (PsMet ` X ) -> ( (pstoMet ` D ) e. ( *Met ` ( X /. .~ ) ) <-> ( (pstoMet ` D ) : ( ( X /. .~ ) X. ( X /. .~ ) ) --> RR* /\ A. x e. ( X /. .~ ) A. y e. ( X /. .~ ) ( ( ( x (pstoMet ` D ) y ) = 0 <-> x = y ) /\ A. z e. ( X /. .~ ) ( x (pstoMet ` D ) y ) <_ ( ( z (pstoMet ` D ) x ) +e ( z (pstoMet ` D ) y ) ) ) ) ) )
96	33, 91, 95	mpbir2and 957	1 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) e. ( *Met ` ( X /. .~ ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   U.cuni 4436   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Er wer 7739   [cec 7740   /.cqs 7741   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730   *Metcxmt 19731  ~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  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-ec 7744  df-qs 7748  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-xmet 19739  df-metid 29931  df-pstm 29932

This theorem is referenced by: (None)