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Theorem metideq 26488
Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metideq  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  =  ( B D F ) )

Proof of Theorem metideq
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  D  e.  (PsMet `  X ) )
2 metidss 26486 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
3 dmss 5150 . . . . . . . . . 10  |-  ( (~Met `  D )  C_  ( X  X.  X )  ->  dom  (~Met `  D )  C_ 
dom  ( X  X.  X ) )
42, 3syl 16 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  dom  (~Met `  D )  C_  dom  ( X  X.  X
) )
5 dmxpid 5170 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
64, 5syl6sseq 3513 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  dom  (~Met `  D )  C_  X
)
71, 6syl 16 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  dom  (~Met `  D )  C_  X
)
8 xpss 5057 . . . . . . . . . . 11  |-  ( X  X.  X )  C_  ( _V  X.  _V )
92, 8syl6ss 3479 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
10 df-rel 4958 . . . . . . . . . 10  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
119, 10sylibr 212 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
121, 11syl 16 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  Rel  (~Met `  D ) )
13 simprl 755 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A (~Met `  D ) B )
14 releldm 5183 . . . . . . . 8  |-  ( ( Rel  (~Met `  D
)  /\  A (~Met `  D ) B )  ->  A  e.  dom  (~Met `  D ) )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A  e.  dom  (~Met `  D )
)
167, 15sseldd 3468 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A  e.  X )
17 simprr 756 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E (~Met `  D ) F )
18 releldm 5183 . . . . . . . 8  |-  ( ( Rel  (~Met `  D
)  /\  E (~Met `  D ) F )  ->  E  e.  dom  (~Met `  D ) )
1912, 17, 18syl2anc 661 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E  e.  dom  (~Met `  D )
)
207, 19sseldd 3468 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E  e.  X )
21 psmetsym 20028 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  E  e.  X )  ->  ( A D E )  =  ( E D A ) )
221, 16, 20, 21syl3anc 1219 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  =  ( E D A ) )
23 psmetf 20024 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2423fovrnda 6347 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  A  e.  X )
)  ->  ( E D A )  e.  RR* )
251, 20, 16, 24syl12anc 1217 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E D A )  e.  RR* )
2622, 25eqeltrd 2542 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  e.  RR* )
27 rnss 5179 . . . . . . . . 9  |-  ( (~Met `  D )  C_  ( X  X.  X )  ->  ran  (~Met `  D )  C_ 
ran  ( X  X.  X ) )
282, 27syl 16 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  ran  (~Met `  D )  C_  ran  ( X  X.  X
) )
29 rnxpid 5382 . . . . . . . 8  |-  ran  ( X  X.  X )  =  X
3028, 29syl6sseq 3513 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ran  (~Met `  D )  C_  X
)
311, 30syl 16 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ran  (~Met `  D )  C_  X
)
32 relelrn 5184 . . . . . . 7  |-  ( ( Rel  (~Met `  D
)  /\  A (~Met `  D ) B )  ->  B  e.  ran  (~Met `  D ) )
3312, 13, 32syl2anc 661 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  B  e.  ran  (~Met `  D )
)
3431, 33sseldd 3468 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  B  e.  X )
3523fovrnda 6347 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( B  e.  X  /\  E  e.  X )
)  ->  ( B D E )  e.  RR* )
361, 34, 20, 35syl12anc 1217 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  e.  RR* )
37 relelrn 5184 . . . . . . . 8  |-  ( ( Rel  (~Met `  D
)  /\  E (~Met `  D ) F )  ->  F  e.  ran  (~Met `  D ) )
3812, 17, 37syl2anc 661 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  F  e.  ran  (~Met `  D )
)
3931, 38sseldd 3468 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  F  e.  X )
40 psmetsym 20028 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  F  e.  X  /\  B  e.  X )  ->  ( F D B )  =  ( B D F ) )
411, 39, 34, 40syl3anc 1219 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D B )  =  ( B D F ) )
4223fovrnda 6347 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( F  e.  X  /\  B  e.  X )
)  ->  ( F D B )  e.  RR* )
431, 39, 34, 42syl12anc 1217 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D B )  e.  RR* )
4441, 43eqeltrrd 2543 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  e.  RR* )
45 psmettri2 20027 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( B  e.  X  /\  A  e.  X  /\  E  e.  X )
)  ->  ( A D E )  <_  (
( B D A ) +e ( B D E ) ) )
461, 34, 16, 20, 45syl13anc 1221 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  (
( B D A ) +e ( B D E ) ) )
47 psmetsym 20028 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
481, 16, 34, 47syl3anc 1219 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D B )  =  ( B D A ) )
4916, 34jca 532 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A  e.  X  /\  B  e.  X ) )
50 metidv 26487 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  ( A D B )  =  0 ) )
5150biimpa 484 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  /\  A (~Met `  D ) B )  ->  ( A D B )  =  0 )
521, 49, 13, 51syl21anc 1218 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D B )  =  0 )
5348, 52eqtr3d 2497 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D A )  =  0 )
5453oveq1d 6218 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( B D A ) +e ( B D E ) )  =  ( 0 +e
( B D E ) ) )
55 xaddid2 11325 . . . . . . 7  |-  ( ( B D E )  e.  RR*  ->  ( 0 +e ( B D E ) )  =  ( B D E ) )
5636, 55syl 16 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( 0 +e ( B D E ) )  =  ( B D E ) )
5754, 56eqtrd 2495 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( B D A ) +e ( B D E ) )  =  ( B D E ) )
5846, 57breqtrd 4427 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  ( B D E ) )
59 psmettri2 20027 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( F  e.  X  /\  B  e.  X  /\  E  e.  X )
)  ->  ( B D E )  <_  (
( F D B ) +e ( F D E ) ) )
601, 39, 34, 20, 59syl13anc 1221 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  <_  (
( F D B ) +e ( F D E ) ) )
61 psmetsym 20028 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  F  e.  X  /\  E  e.  X )  ->  ( F D E )  =  ( E D F ) )
621, 39, 20, 61syl3anc 1219 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D E )  =  ( E D F ) )
6320, 39jca 532 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E  e.  X  /\  F  e.  X ) )
64 metidv 26487 . . . . . . . . . 10  |-  ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  F  e.  X )
)  ->  ( E
(~Met `  D ) F 
<->  ( E D F )  =  0 ) )
6564biimpa 484 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  F  e.  X )
)  /\  E (~Met `  D ) F )  ->  ( E D F )  =  0 )
661, 63, 17, 65syl21anc 1218 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E D F )  =  0 )
6762, 66eqtrd 2495 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D E )  =  0 )
6867oveq2d 6219 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( F D B ) +e ( F D E ) )  =  ( ( F D B ) +e 0 ) )
69 xaddid1 11324 . . . . . . 7  |-  ( ( F D B )  e.  RR*  ->  ( ( F D B ) +e 0 )  =  ( F D B ) )
7043, 69syl 16 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( F D B ) +e 0 )  =  ( F D B ) )
7168, 70, 413eqtrd 2499 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( F D B ) +e ( F D E ) )  =  ( B D F ) )
7260, 71breqtrd 4427 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  <_  ( B D F ) )
7326, 36, 44, 58, 72xrletrd 11251 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  ( B D F ) )
7423fovrnda 6347 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  F  e.  X )
)  ->  ( A D F )  e.  RR* )
751, 16, 39, 74syl12anc 1217 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  e.  RR* )
76 psmettri2 20027 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  F  e.  X )
)  ->  ( B D F )  <_  (
( A D B ) +e ( A D F ) ) )
771, 16, 34, 39, 76syl13anc 1221 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  (
( A D B ) +e ( A D F ) ) )
7852oveq1d 6218 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( A D B ) +e ( A D F ) )  =  ( 0 +e
( A D F ) ) )
79 xaddid2 11325 . . . . . . 7  |-  ( ( A D F )  e.  RR*  ->  ( 0 +e ( A D F ) )  =  ( A D F ) )
8075, 79syl 16 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( 0 +e ( A D F ) )  =  ( A D F ) )
8178, 80eqtrd 2495 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( A D B ) +e ( A D F ) )  =  ( A D F ) )
8277, 81breqtrd 4427 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  ( A D F ) )
83 psmettri2 20027 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  A  e.  X  /\  F  e.  X )
)  ->  ( A D F )  <_  (
( E D A ) +e ( E D F ) ) )
841, 20, 16, 39, 83syl13anc 1221 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  <_  (
( E D A ) +e ( E D F ) ) )
85 xaddid1 11324 . . . . . . 7  |-  ( ( E D A )  e.  RR*  ->  ( ( E D A ) +e 0 )  =  ( E D A ) )
8625, 85syl 16 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( E D A ) +e 0 )  =  ( E D A ) )
8766oveq2d 6219 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( E D A ) +e ( E D F ) )  =  ( ( E D A ) +e 0 ) )
8886, 87, 223eqtr4d 2505 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( E D A ) +e ( E D F ) )  =  ( A D E ) )
8984, 88breqtrd 4427 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  <_  ( A D E ) )
9044, 75, 26, 82, 89xrletrd 11251 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  ( A D E ) )
9173, 90jca 532 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( A D E )  <_ 
( B D F )  /\  ( B D F )  <_ 
( A D E ) ) )
92 xrletri3 11244 . . 3  |-  ( ( ( A D E )  e.  RR*  /\  ( B D F )  e. 
RR* )  ->  (
( A D E )  =  ( B D F )  <->  ( ( A D E )  <_ 
( B D F )  /\  ( B D F )  <_ 
( A D E ) ) ) )
9326, 44, 92syl2anc 661 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( A D E )  =  ( B D F )  <->  ( ( A D E )  <_ 
( B D F )  /\  ( B D F )  <_ 
( A D E ) ) ) )
9491, 93mpbird 232 1  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  =  ( B D F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   class class class wbr 4403    X. cxp 4949   dom cdm 4951   ran crn 4952   Rel wrel 4956   ` cfv 5529  (class class class)co 6203   0cc0 9397   RR*cxr 9532    <_ cle 9534   +ecxad 11202  PsMetcpsmet 17935  ~Metcmetid 26481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-xadd 11205  df-psmet 17944  df-metid 26483
This theorem is referenced by:  pstmfval  26491
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