Theorem setsmsdsg 15581

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 13571	. . . 4 |- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
4		9re 9391	. . . . . 6 |- 9 e. RR
5		1nn 9315	. . . . . . 7 |- 1 e. NN
6		2nn0 9580	. . . . . . 7 |- 2 e. NN0
7		9nn0 9587	. . . . . . 7 |- 9 e. NN0
8		9lt10 9907	. . . . . . 7 |- 9 < ; 1 0
9	5, 6, 7, 8	declti 9814	. . . . . 6 |- 9 < ; 1 2
10	4, 9	gtneii 8421	. . . . 5 |- ; 1 2 =/= 9
11		dsndx 13569	. . . . . 6 |- ( dist ` ndx ) = ; 1 2
12		tsetndx 13540	. . . . . 6 |- (TopSet ` ndx ) = 9
13	11, 12	neeq12i 2437	. . . . 5 |- ( ( dist ` ndx ) =/= (TopSet ` ndx ) <-> ; 1 2 =/= 9 )
14	10, 13	mpbir 146	. . . 4 |- ( dist ` ndx ) =/= (TopSet ` ndx )
15		tsetslid 13542	. . . . 5 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
16	15	simpri 113	. . . 4 |- (TopSet ` ndx ) e. NN
17	3, 14, 16	setsslnid 13404	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( dist ` M ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	1, 2, 17	syl2anc 415	. 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 5699	. 2 |- ( ph -> ( dist ` K ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
21	18, 20	eqtr4d 2274	1 |- ( ph -> ( dist ` M ) = ( dist ` K ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   <.cop 3712   X. cxp 4772   |` cres 4776   ` cfv 5377  (class class class)co 6085   1c1 8180   NNcn 9304   2c2 9355   9c9 9362  ;cdc 9777   ndxcnx 13349   sSet csts 13350  Slot cslot 13351   Basecbs 13352  TopSetcts 13437   distcds 13440   MetOpencmopn 14878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-ndx 13355  df-slot 13356  df-sets 13359  df-tset 13450  df-ds 13453

This theorem is used by: (None)