Theorem xmspropd 15064

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 2242	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	3, 4	tpspropd 14623	. . 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 4719	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
8	7	reseq2d 4978	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	6, 8	eqtr3d 2242	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10	2	sqxpeqd 4719	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
11	10	reseq2d 4978	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	9, 11	eqtr3d 2242	. . . . 5 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
13	12	fveq2d 5603	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
14	4, 13	eqeq12d 2222	. . 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 2207	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2207	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2207	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isxms 15038	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) )
20		eqid 2207	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2207	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2207	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isxms 15038	. 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 1373   e. wcel 2178   X. cxp 4691   |` cres 4695   ` cfv 5290   Basecbs 12947   distcds 13033   TopOpenctopn 13187   MetOpencmopn 14418   TopSpctps 14617   *MetSpcxms 14923

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-ndx 12950  df-slot 12951  df-base 12953  df-tset 13043  df-rest 13188  df-topn 13189  df-top 14585  df-topon 14598  df-topsp 14618  df-xms 14926

This theorem is referenced by:  mspropd  15065