Theorem setsmsdsg 15291

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 13380	. . . 4 |- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
4		9re 9289	. . . . . 6 |- 9 e. RR
5		1nn 9213	. . . . . . 7 |- 1 e. NN
6		2nn0 9478	. . . . . . 7 |- 2 e. NN0
7		9nn0 9485	. . . . . . 7 |- 9 e. NN0
8		9lt10 9802	. . . . . . 7 |- 9 < ; 1 0
9	5, 6, 7, 8	declti 9709	. . . . . 6 |- 9 < ; 1 2
10	4, 9	gtneii 8334	. . . . 5 |- ; 1 2 =/= 9
11		dsndx 13378	. . . . . 6 |- ( dist ` ndx ) = ; 1 2
12		tsetndx 13349	. . . . . 6 |- (TopSet ` ndx ) = 9
13	11, 12	neeq12i 2420	. . . . 5 |- ( ( dist ` ndx ) =/= (TopSet ` ndx ) <-> ; 1 2 =/= 9 )
14	10, 13	mpbir 146	. . . 4 |- ( dist ` ndx ) =/= (TopSet ` ndx )
15		tsetslid 13351	. . . . 5 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
16	15	simpri 113	. . . 4 |- (TopSet ` ndx ) e. NN
17	3, 14, 16	setsslnid 13214	. . 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 5652	. 2 |- ( ph -> ( dist ` K ) = ( dist ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
21	18, 20	eqtr4d 2267	1 |- ( ph -> ( dist ` M ) = ( dist ` K ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   <.cop 3676   X. cxp 4729   |` cres 4733   ` cfv 5333  (class class class)co 6028   1c1 8093   NNcn 9202   2c2 9253   9c9 9260  ;cdc 9672   ndxcnx 13159   sSet csts 13160  Slot cslot 13161   Basecbs 13162  TopSetcts 13246   distcds 13249   MetOpencmopn 14637

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-ndx 13165  df-slot 13166  df-sets 13169  df-tset 13259  df-ds 13262

This theorem is referenced by: (None)