Theorem xmspropd 15151

Description: Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
xmspropd.1	|- ( ph -> B = ( Base ` K ) )
xmspropd.2	|- ( ph -> B = ( Base ` L ) )
xmspropd.3	|- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )
xmspropd.4	|- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Assertion

Ref	Expression
xmspropd	|- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )

Proof of Theorem xmspropd

Step	Hyp	Ref	Expression
1		xmspropd.1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
2		xmspropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
3	1, 2	eqtr3d 2264	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	3, 4	tpspropd 14710	. . 3 |- ( ph -> ( K e. TopSp <-> L e. TopSp ) )
6		xmspropd.3	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )
7	1	sqxpeqd 4745	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
8	7	reseq2d 5005	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	6, 8	eqtr3d 2264	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10	2	sqxpeqd 4745	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
11	10	reseq2d 5005	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	9, 11	eqtr3d 2264	. . . . 5 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
13	12	fveq2d 5631	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
14	4, 13	eqeq12d 2244	. . 3 |- ( ph -> ( ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) <-> ( TopOpen ` L ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) ) )
15	5, 14	anbi12d 473	. 2 |- ( ph -> ( ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) <-> ( L e. TopSp /\ ( TopOpen ` L ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) ) ) )
16		eqid 2229	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2229	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2229	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isxms 15125	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) )
20		eqid 2229	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2229	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2229	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isxms 15125	. 2 |- ( L e. *MetSp <-> ( L e. TopSp /\ ( TopOpen ` L ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) ) )
24	15, 19, 23	3bitr4g 223	1 |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   X. cxp 4717   |` cres 4721   ` cfv 5318   Basecbs 13032   distcds 13119   TopOpenctopn 13273   MetOpencmopn 14505   TopSpctps 14704   *MetSpcxms 15010

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-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  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-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  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-iun 3967  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-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-ndx 13035  df-slot 13036  df-base 13038  df-tset 13129  df-rest 13274  df-topn 13275  df-top 14672  df-topon 14685  df-topsp 14705  df-xms 15013

This theorem is referenced by:  mspropd  15152