Theorem mspropd 15201

Description: Property deduction for a 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
mspropd	|- ( ph -> ( K e. MetSp <-> L e. MetSp ) )

Proof of Theorem mspropd

Step	Hyp	Ref	Expression
1		xmspropd.1	. . . 4 |- ( ph -> B = ( Base ` K ) )
2		xmspropd.2	. . . 4 |- ( ph -> B = ( Base ` L ) )
3		xmspropd.3	. . . 4 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	1, 2, 3, 4	xmspropd 15200	. . 3 |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )
6	1	sqxpeqd 4751	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
7	6	reseq2d 5013	. . . . . 6 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
8	3, 7	eqtr3d 2266	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	2	sqxpeqd 4751	. . . . . 6 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
10	9	reseq2d 5013	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
11	8, 10	eqtr3d 2266	. . . 4 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	1, 2	eqtr3d 2266	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
13	12	fveq2d 5643	. . . 4 |- ( ph -> ( Met ` ( Base ` K ) ) = ( Met ` ( Base ` L ) ) )
14	11, 13	eleq12d 2302	. . 3 |- ( ph -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) <-> ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( Base ` L ) ) ) )
15	5, 14	anbi12d 473	. 2 |- ( ph -> ( ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) <-> ( L e. *MetSp /\ ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( 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	isms 15176	. 2 |- ( K e. MetSp <-> ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( 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	isms 15176	. 2 |- ( L e. MetSp <-> ( L e. *MetSp /\ ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( 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 1397   e. wcel 2202   X. cxp 4723   |` cres 4727   ` cfv 5326   Basecbs 13081   distcds 13168   TopOpenctopn 13322   Metcmet 14550   *MetSpcxms 15059   MetSpcms 15060

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-ndx 13084  df-slot 13085  df-base 13087  df-tset 13178  df-rest 13323  df-topn 13324  df-top 14721  df-topon 14734  df-topsp 14754  df-xms 15062  df-ms 15063

This theorem is referenced by: (None)