Theorem xmspropd 15288

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 2266	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	3, 4	tpspropd 14847	. . 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 4757	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
8	7	reseq2d 5019	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	6, 8	eqtr3d 2266	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10	2	sqxpeqd 4757	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
11	10	reseq2d 5019	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	9, 11	eqtr3d 2266	. . . . 5 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
13	12	fveq2d 5652	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
14	4, 13	eqeq12d 2246	. . 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 2231	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2231	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2231	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isxms 15262	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) )
20		eqid 2231	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2231	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2231	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isxms 15262	. 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 2202   X. cxp 4729   |` cres 4733   ` cfv 5333   Basecbs 13162   distcds 13249   TopOpenctopn 13403   MetOpencmopn 14637   TopSpctps 14841   *MetSpcxms 15147

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-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  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-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  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-iun 3977  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-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  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-ndx 13165  df-slot 13166  df-base 13168  df-tset 13259  df-rest 13404  df-topn 13405  df-top 14809  df-topon 14822  df-topsp 14842  df-xms 15150

This theorem is referenced by:  mspropd  15289