Theorem setsmsdsg 15706

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 13630	. . . 4 |- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
4		9re 9400	. . . . . 6 |- 9 e. RR
5		1nn 9324	. . . . . . 7 |- 1 e. NN
6		2nn0 9591	. . . . . . 7 |- 2 e. NN0
7		9nn0 9598	. . . . . . 7 |- 9 e. NN0
8		9lt10 9923	. . . . . . 7 |- 9 < ; 1 0
9	5, 6, 7, 8	declti 9830	. . . . . 6 |- 9 < ; 1 2
10	4, 9	gtneii 8428	. . . . 5 |- ; 1 2 =/= 9
11		dsndx 13628	. . . . . 6 |- ( dist ` ndx ) = ; 1 2
12		tsetndx 13599	. . . . . 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 13601	. . . . 5 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
16	15	simpri 113	. . . 4 |- (TopSet ` ndx ) e. NN
17	3, 14, 16	setsslnid 13462	. . 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 5702	. 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 4774   |` cres 4778   ` cfv 5380  (class class class)co 6088   1c1 8186   NNcn 9313   2c2 9364   9c9 9371  ;cdc 9788   ndxcnx 13407   sSet csts 13408  Slot cslot 13409   Basecbs 13410  TopSetcts 13496   distcds 13499   MetOpencmopn 14987

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-ndx 13413  df-slot 13414  df-sets 13417  df-tset 13509  df-ds 13512

This theorem is used by: (None)