Theorem setsmsdsg 15169

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

Step	Hyp	Ref	Expression
1		setsmsbasg.m	. . 3 |- ( ph -> M e. V )
2		setsmsbasg.d	. . 3 |- ( ph -> ( MetOpen ` D ) e. W )
3		dsslid 13265	. . . 4 |- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
4		9re 9208	. . . . . 6 |- 9 e. RR
5		1nn 9132	. . . . . . 7 |- 1 e. NN
6		2nn0 9397	. . . . . . 7 |- 2 e. NN0
7		9nn0 9404	. . . . . . 7 |- 9 e. NN0
8		9lt10 9719	. . . . . . 7 |- 9 < ; 1 0
9	5, 6, 7, 8	declti 9626	. . . . . 6 |- 9 < ; 1 2
10	4, 9	gtneii 8253	. . . . 5 |- ; 1 2 =/= 9
11		dsndx 13263	. . . . . 6 |- ( dist ` ndx ) = ; 1 2
12		tsetndx 13234	. . . . . 6 |- (TopSet ` ndx ) = 9
13	11, 12	neeq12i 2417	. . . . 5 |- ( ( dist ` ndx ) =/= (TopSet ` ndx ) <-> ; 1 2 =/= 9 )
14	10, 13	mpbir 146	. . . 4 |- ( dist ` ndx ) =/= (TopSet ` ndx )
15		tsetslid 13236	. . . . 5 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
16	15	simpri 113	. . . 4 |- (TopSet ` ndx ) e. NN
17	3, 14, 16	setsslnid 13099	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( dist ` M ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	1, 2, 17	syl2anc 411	. 2 |- ( ph -> ( dist ` M ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
19		setsms.k	. . 3 |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
20	19	fveq2d 5633	. 2 |- ( ph -> ( dist ` K ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
21	18, 20	eqtr4d 2265	1 |- ( ph -> ( dist ` M ) = ( dist ` K ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   <.cop 3669   X. cxp 4717   |` cres 4721   ` cfv 5318  (class class class)co 6007   1c1 8011   NNcn 9121   2c2 9172   9c9 9179  ;cdc 9589   ndxcnx 13044   sSet csts 13045  Slot cslot 13046   Basecbs 13047  TopSetcts 13131   distcds 13134   MetOpencmopn 14520

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-ndx 13050  df-slot 13051  df-sets 13054  df-tset 13144  df-ds 13147

This theorem is referenced by: (None)