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Theorem metideq 28205
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 455 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  D  e.  (PsMet `  X ) )
2 metidss 28203 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( X  X.  X ) )
3 dmss 5144 . . . . . . . . 9  |-  ( (~Met `  D )  C_  ( X  X.  X )  ->  dom  (~Met `  D )  C_ 
dom  ( X  X.  X ) )
42, 3syl 17 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  dom  (~Met `  D )  C_  dom  ( X  X.  X
) )
5 dmxpid 5164 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
64, 5syl6sseq 3487 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  dom  (~Met `  D )  C_  X
)
71, 6syl 17 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  dom  (~Met `  D )  C_  X
)
8 xpss 5051 . . . . . . . . . 10  |-  ( X  X.  X )  C_  ( _V  X.  _V )
92, 8syl6ss 3453 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  (~Met `  D
)  C_  ( _V  X.  _V ) )
10 df-rel 4949 . . . . . . . . 9  |-  ( Rel  (~Met `  D )  <->  (~Met `  D )  C_  ( _V  X.  _V ) )
119, 10sylibr 212 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  Rel  (~Met `  D ) )
121, 11syl 17 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  Rel  (~Met `  D ) )
13 simprl 756 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A (~Met `  D ) B )
14 releldm 5177 . . . . . . 7  |-  ( ( Rel  (~Met `  D
)  /\  A (~Met `  D ) B )  ->  A  e.  dom  (~Met `  D ) )
1512, 13, 14syl2anc 659 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A  e.  dom  (~Met `  D )
)
167, 15sseldd 3442 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  A  e.  X )
17 simprr 758 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E (~Met `  D ) F )
18 releldm 5177 . . . . . . 7  |-  ( ( Rel  (~Met `  D
)  /\  E (~Met `  D ) F )  ->  E  e.  dom  (~Met `  D ) )
1912, 17, 18syl2anc 659 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E  e.  dom  (~Met `  D )
)
207, 19sseldd 3442 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  E  e.  X )
21 psmetsym 20998 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  E  e.  X )  ->  ( A D E )  =  ( E D A ) )
221, 16, 20, 21syl3anc 1230 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  =  ( E D A ) )
23 psmetf 20994 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2423fovrnda 6383 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  A  e.  X )
)  ->  ( E D A )  e.  RR* )
251, 20, 16, 24syl12anc 1228 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E D A )  e.  RR* )
2622, 25eqeltrd 2490 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  e.  RR* )
27 rnss 5173 . . . . . . . 8  |-  ( (~Met `  D )  C_  ( X  X.  X )  ->  ran  (~Met `  D )  C_ 
ran  ( X  X.  X ) )
282, 27syl 17 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  ran  (~Met `  D )  C_  ran  ( X  X.  X
) )
29 rnxpid 5379 . . . . . . 7  |-  ran  ( X  X.  X )  =  X
3028, 29syl6sseq 3487 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ran  (~Met `  D )  C_  X
)
311, 30syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ran  (~Met `  D )  C_  X
)
32 relelrn 5178 . . . . . 6  |-  ( ( Rel  (~Met `  D
)  /\  A (~Met `  D ) B )  ->  B  e.  ran  (~Met `  D ) )
3312, 13, 32syl2anc 659 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  B  e.  ran  (~Met `  D )
)
3431, 33sseldd 3442 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  B  e.  X )
3523fovrnda 6383 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( B  e.  X  /\  E  e.  X )
)  ->  ( B D E )  e.  RR* )
361, 34, 20, 35syl12anc 1228 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  e.  RR* )
37 relelrn 5178 . . . . . . 7  |-  ( ( Rel  (~Met `  D
)  /\  E (~Met `  D ) F )  ->  F  e.  ran  (~Met `  D ) )
3812, 17, 37syl2anc 659 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  F  e.  ran  (~Met `  D )
)
3931, 38sseldd 3442 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  F  e.  X )
40 psmetsym 20998 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  F  e.  X  /\  B  e.  X )  ->  ( F D B )  =  ( B D F ) )
411, 39, 34, 40syl3anc 1230 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D B )  =  ( B D F ) )
4223fovrnda 6383 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( F  e.  X  /\  B  e.  X )
)  ->  ( F D B )  e.  RR* )
431, 39, 34, 42syl12anc 1228 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D B )  e.  RR* )
4441, 43eqeltrrd 2491 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  e.  RR* )
45 psmettri2 20997 . . . . 5  |-  ( ( 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 1232 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  (
( B D A ) +e ( B D E ) ) )
47 psmetsym 20998 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
481, 16, 34, 47syl3anc 1230 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D B )  =  ( B D A ) )
4916, 34jca 530 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A  e.  X  /\  B  e.  X ) )
50 metidv 28204 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A
(~Met `  D ) B 
<->  ( A D B )  =  0 ) )
5150biimpa 482 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
)  /\  A (~Met `  D ) B )  ->  ( A D B )  =  0 )
521, 49, 13, 51syl21anc 1229 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D B )  =  0 )
5348, 52eqtr3d 2445 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D A )  =  0 )
5453oveq1d 6249 . . . . 5  |-  ( ( 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 11410 . . . . . 6  |-  ( ( B D E )  e.  RR*  ->  ( 0 +e ( B D E ) )  =  ( B D E ) )
5636, 55syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( 0 +e ( B D E ) )  =  ( B D E ) )
5754, 56eqtrd 2443 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( B D A ) +e ( B D E ) )  =  ( B D E ) )
5846, 57breqtrd 4418 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  ( B D E ) )
59 psmettri2 20997 . . . . 5  |-  ( ( 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 1232 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  <_  (
( F D B ) +e ( F D E ) ) )
61 psmetsym 20998 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  F  e.  X  /\  E  e.  X )  ->  ( F D E )  =  ( E D F ) )
621, 39, 20, 61syl3anc 1230 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D E )  =  ( E D F ) )
6320, 39jca 530 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E  e.  X  /\  F  e.  X ) )
64 metidv 28204 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  F  e.  X )
)  ->  ( E
(~Met `  D ) F 
<->  ( E D F )  =  0 ) )
6564biimpa 482 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  ( E  e.  X  /\  F  e.  X )
)  /\  E (~Met `  D ) F )  ->  ( E D F )  =  0 )
661, 63, 17, 65syl21anc 1229 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( E D F )  =  0 )
6762, 66eqtrd 2443 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( F D E )  =  0 )
6867oveq2d 6250 . . . . 5  |-  ( ( 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 11409 . . . . . 6  |-  ( ( F D B )  e.  RR*  ->  ( ( F D B ) +e 0 )  =  ( F D B ) )
7043, 69syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( F D B ) +e 0 )  =  ( F D B ) )
7168, 70, 413eqtrd 2447 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( F D B ) +e ( F D E ) )  =  ( B D F ) )
7260, 71breqtrd 4418 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D E )  <_  ( B D F ) )
7326, 36, 44, 58, 72xrletrd 11336 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  <_  ( B D F ) )
7423fovrnda 6383 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  F  e.  X )
)  ->  ( A D F )  e.  RR* )
751, 16, 39, 74syl12anc 1228 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  e.  RR* )
76 psmettri2 20997 . . . . 5  |-  ( ( 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 1232 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  (
( A D B ) +e ( A D F ) ) )
7852oveq1d 6249 . . . . 5  |-  ( ( 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 11410 . . . . . 6  |-  ( ( A D F )  e.  RR*  ->  ( 0 +e ( A D F ) )  =  ( A D F ) )
8075, 79syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( 0 +e ( A D F ) )  =  ( A D F ) )
8178, 80eqtrd 2443 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( A D B ) +e ( A D F ) )  =  ( A D F ) )
8277, 81breqtrd 4418 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  ( A D F ) )
83 psmettri2 20997 . . . . 5  |-  ( ( 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 1232 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  <_  (
( E D A ) +e ( E D F ) ) )
85 xaddid1 11409 . . . . . 6  |-  ( ( E D A )  e.  RR*  ->  ( ( E D A ) +e 0 )  =  ( E D A ) )
8625, 85syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( E D A ) +e 0 )  =  ( E D A ) )
8766oveq2d 6250 . . . . 5  |-  ( ( 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 2453 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( ( E D A ) +e ( E D F ) )  =  ( A D E ) )
8984, 88breqtrd 4418 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D F )  <_  ( A D E ) )
9044, 75, 26, 82, 89xrletrd 11336 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( B D F )  <_  ( A D E ) )
91 xrletri3 11329 . . 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 ) ) ) )
9226, 44, 91syl2anc 659 . 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 ) ) ) )
9373, 90, 92mpbir2and 923 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    X. cxp 4940   dom cdm 4942   ran crn 4943   Rel wrel 4947   ` cfv 5525  (class class class)co 6234   0cc0 9442   RR*cxr 9577    <_ cle 9579   +ecxad 11287  PsMetcpsmet 18614  ~Metcmetid 28198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-xadd 11290  df-psmet 18623  df-metid 28200
This theorem is referenced by:  pstmfval  28208
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