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Theorem psmetres2 22119
Description: Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmetres2 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. (PsMet ` R ) )
Proof of Theorem psmetres2
Dummy variables a b c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psmetf 22111 . . . 4 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
21adantr 481 . . 3 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> D : ( X X. X ) --> RR* )
3 simpr 477 . . . 4 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> R C_ X )
4 xpss12 5225 . . . 4 |- ( ( R C_ X /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
53, 3, 4syl2anc 693 . . 3 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
62, 5fssresd 6071 . 2 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
7 simpr 477 . . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> a e. R )
87, 7ovresd 6801 . . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> ( a ( D |` ( R X. R ) ) a ) = ( a D a ) )
9 simpll 790 . . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> D e. (PsMet ` X ) )
103sselda 3603 . . . . . 6 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> a e. X )
11 psmet0 22113 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ a e. X ) -> ( a D a ) = 0 )
129, 10, 11syl2anc 693 . . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> ( a D a ) = 0 )
138, 12eqtrd 2656 . . . 4 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> ( a ( D |` ( R X. R ) ) a ) = 0 )
149ad2antrr 762 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> D e. (PsMet ` X ) )
153ad2antrr 762 . . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> R C_ X )
1615sselda 3603 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> c e. X )
1710ad2antrr 762 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> a e. X )
183adantr 481 . . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> R C_ X )
1918sselda 3603 . . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> b e. X )
2019adantr 481 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> b e. X )
21 psmettri2 22114 . . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( c e. X /\ a e. X /\ b e. X ) ) -> ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) )
2214, 16, 17, 20, 21syl13anc 1328 . . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( a D b ) <_ ( ( c D a ) +e ( c D b ) ) )
237adantr 481 . . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> a e. R )
24 simpr 477 . . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> b e. R )
2523, 24ovresd 6801 . . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> ( a ( D |` ( R X. R ) ) b ) = ( a D b ) )
2625adantr 481 . . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( a ( D |` ( R X. R ) ) b ) = ( a D b ) )
27 simpr 477 . . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> c e. R )
287ad2antrr 762 . . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> a e. R )
2927, 28ovresd 6801 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( c ( D |` ( R X. R ) ) a ) = ( c D a ) )
3024adantr 481 . . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> b e. R )
3127, 30ovresd 6801 . . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( c ( D |` ( R X. R ) ) b ) = ( c D b ) )
3229, 31oveq12d 6668 . . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) = ( ( c D a ) +e ( c D b ) ) )
3322, 26, 323brtr4d 4685 . . . . . 6 |- ( ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) /\ c e. R ) -> ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) )
3433ralrimiva 2966 . . . . 5 |- ( ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) /\ b e. R ) -> A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) )
3534ralrimiva 2966 . . . 4 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> A. b e. R A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) )
3613, 35jca 554 . . 3 |- ( ( ( D e. (PsMet ` X ) /\ R C_ X ) /\ a e. R ) -> ( ( a ( D |` ( R X. R ) ) a ) = 0 /\ A. b e. R A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) ) )
3736ralrimiva 2966 . 2 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> A. a e. R ( ( a ( D |` ( R X. R ) ) a ) = 0 /\ A. b e. R A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) ) )
38 elfvex 6221 . . . . 5 |- ( D e. (PsMet ` X ) -> X e. _V )
3938adantr 481 . . . 4 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> X e. _V )
4039, 3ssexd 4805 . . 3 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> R e. _V )
41 ispsmet 22109 . . 3 |- ( R e. _V -> ( ( D |` ( R X. R ) ) e. (PsMet ` R ) <-> ( ( D |` ( R X. R ) ) : ( R X. R ) --> RR* /\ A. a e. R ( ( a ( D |` ( R X. R ) ) a ) = 0 /\ A. b e. R A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) ) ) ) )
4240, 41syl 17 . 2 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( ( D |` ( R X. R ) ) e. (PsMet ` R ) <-> ( ( D |` ( R X. R ) ) : ( R X. R ) --> RR* /\ A. a e. R ( ( a ( D |` ( R X. R ) ) a ) = 0 /\ A. b e. R A. c e. R ( a ( D |` ( R X. R ) ) b ) <_ ( ( c ( D |` ( R X. R ) ) a ) +e ( c ( D |` ( R X. R ) ) b ) ) ) ) ) )
436, 37, 42mpbir2and 957 1 |- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. (PsMet ` R ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730
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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-psmet 19738
This theorem is referenced by:  restmetu  22375
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