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Theorem setsmsdsg 13613
Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
setsms.x  |-  ( ph  ->  X  =  ( Base `  M ) )
setsms.d  |-  ( ph  ->  D  =  ( (
dist `  M )  |`  ( X  X.  X
) ) )
setsms.k  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
setsmsbasg.m  |-  ( ph  ->  M  e.  V )
setsmsbasg.d  |-  ( ph  ->  ( MetOpen `  D )  e.  W )
Assertion
Ref Expression
setsmsdsg  |-  ( ph  ->  ( dist `  M
)  =  ( dist `  K ) )

Proof of Theorem setsmsdsg
StepHypRef Expression
1 setsmsbasg.m . . 3  |-  ( ph  ->  M  e.  V )
2 setsmsbasg.d . . 3  |-  ( ph  ->  ( MetOpen `  D )  e.  W )
3 dsslid 12614 . . . 4  |-  ( dist 
= Slot  ( dist `  ndx )  /\  ( dist `  ndx )  e.  NN )
4 9re 8982 . . . . . 6  |-  9  e.  RR
5 1nn 8906 . . . . . . 7  |-  1  e.  NN
6 2nn0 9169 . . . . . . 7  |-  2  e.  NN0
7 9nn0 9176 . . . . . . 7  |-  9  e.  NN0
8 9lt10 9490 . . . . . . 7  |-  9  < ; 1
0
95, 6, 7, 8declti 9397 . . . . . 6  |-  9  < ; 1
2
104, 9gtneii 8030 . . . . 5  |- ; 1 2  =/=  9
11 dsndx 12612 . . . . . 6  |-  ( dist `  ndx )  = ; 1 2
12 tsetndx 12595 . . . . . 6  |-  (TopSet `  ndx )  =  9
1311, 12neeq12i 2364 . . . . 5  |-  ( (
dist `  ndx )  =/=  (TopSet `  ndx )  <-> ; 1 2  =/=  9
)
1410, 13mpbir 146 . . . 4  |-  ( dist `  ndx )  =/=  (TopSet ` 
ndx )
15 tsetslid 12597 . . . . 5  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
1615simpri 113 . . . 4  |-  (TopSet `  ndx )  e.  NN
173, 14, 16setsslnid 12483 . . 3  |-  ( ( M  e.  V  /\  ( MetOpen `  D )  e.  W )  ->  ( dist `  M )  =  ( dist `  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
) )
181, 2, 17syl2anc 411 . 2  |-  ( ph  ->  ( dist `  M
)  =  ( dist `  ( M sSet  <. (TopSet ` 
ndx ) ,  (
MetOpen `  D ) >.
) ) )
19 setsms.k . . 3  |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
2019fveq2d 5514 . 2  |-  ( ph  ->  ( dist `  K
)  =  ( dist `  ( M sSet  <. (TopSet ` 
ndx ) ,  (
MetOpen `  D ) >.
) ) )
2118, 20eqtr4d 2213 1  |-  ( ph  ->  ( dist `  M
)  =  ( dist `  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    =/= wne 2347   <.cop 3594    X. cxp 4620    |` cres 4624   ` cfv 5211  (class class class)co 5868   1c1 7790   NNcn 8895   2c2 8946   9c9 8953  ;cdc 9360   ndxcnx 12429   sSet csts 12430  Slot cslot 12431   Basecbs 12432  TopSetcts 12511   distcds 12514   MetOpencmopn 13118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905  ax-pre-mulgt0 7906
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-5 8957  df-6 8958  df-7 8959  df-8 8960  df-9 8961  df-n0 9153  df-z 9230  df-dec 9361  df-ndx 12435  df-slot 12436  df-sets 12439  df-tset 12524  df-ds 12527
This theorem is referenced by: (None)
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