Theorem setsmsdsg 13274

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 12578	. . . 4 |- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
4		9re 8965	. . . . . 6 |- 9 e. RR
5		1nn 8889	. . . . . . 7 |- 1 e. NN
6		2nn0 9152	. . . . . . 7 |- 2 e. NN0
7		9nn0 9159	. . . . . . 7 |- 9 e. NN0
8		9lt10 9473	. . . . . . 7 |- 9 < ; 1 0
9	5, 6, 7, 8	declti 9380	. . . . . 6 |- 9 < ; 1 2
10	4, 9	gtneii 8015	. . . . 5 |- ; 1 2 =/= 9
11		dsndx 12576	. . . . . 6 |- ( dist ` ndx ) = ; 1 2
12		tsetndx 12566	. . . . . 6 |- (TopSet ` ndx ) = 9
13	11, 12	neeq12i 2357	. . . . 5 |- ( ( dist ` ndx ) =/= (TopSet ` ndx ) <-> ; 1 2 =/= 9 )
14	10, 13	mpbir 145	. . . 4 |- ( dist ` ndx ) =/= (TopSet ` ndx )
15		tsetslid 12568	. . . . 5 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
16	15	simpri 112	. . . 4 |- (TopSet ` ndx ) e. NN
17	3, 14, 16	setsslnid 12467	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( dist ` M ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	1, 2, 17	syl2anc 409	. 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 5500	. 2 |- ( ph -> ( dist ` K ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
21	18, 20	eqtr4d 2206	1 |- ( ph -> ( dist ` M ) = ( dist ` K ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   =/= wne 2340   <.cop 3586   X. cxp 4609   |` cres 4613   ` cfv 5198  (class class class)co 5853   1c1 7775   NNcn 8878   2c2 8929   9c9 8936  ;cdc 9343   ndxcnx 12413   sSet csts 12414  Slot cslot 12415   Basecbs 12416  TopSetcts 12486   distcds 12489   MetOpencmopn 12779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890  ax-pre-mulgt0 7891

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-n0 9136  df-z 9213  df-dec 9344  df-ndx 12419  df-slot 12420  df-sets 12423  df-tset 12499  df-ds 12502

This theorem is referenced by: (None)