Theorem xmspropd 15359

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 2269	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	3, 4	tpspropd 14918	. . 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 4777	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
8	7	reseq2d 5040	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	6, 8	eqtr3d 2269	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10	2	sqxpeqd 4777	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
11	10	reseq2d 5040	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	9, 11	eqtr3d 2269	. . . . 5 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
13	12	fveq2d 5676	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
14	4, 13	eqeq12d 2249	. . 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 2234	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2234	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2234	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isxms 15333	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) )
20		eqid 2234	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2234	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2234	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isxms 15333	. 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 1398   e. wcel 2205   X. cxp 4749   |` cres 4753   ` cfv 5354   Basecbs 13229   distcds 13316   TopOpenctopn 13470   MetOpencmopn 14706   TopSpctps 14912   *MetSpcxms 15218

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-ndx 13232  df-slot 13233  df-base 13235  df-tset 13326  df-rest 13471  df-topn 13472  df-top 14880  df-topon 14893  df-topsp 14913  df-xms 15221

This theorem is referenced by:  mspropd  15360