Theorem mspropd 15579

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 15578	. . 3 |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )
6	1	sqxpeqd 4800	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
7	6	reseq2d 5063	. . . . . 6 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
8	3, 7	eqtr3d 2273	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	2	sqxpeqd 4800	. . . . . 6 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
10	9	reseq2d 5063	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
11	8, 10	eqtr3d 2273	. . . 4 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	1, 2	eqtr3d 2273	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
13	12	fveq2d 5699	. . . 4 |- ( ph -> ( Met ` ( Base ` K ) ) = ( Met ` ( Base ` L ) ) )
14	11, 13	eleq12d 2309	. . 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 477	. 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 2238	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2238	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2238	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isms 15554	. 2 |- ( K e. MetSp <-> ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) )
20		eqid 2238	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2238	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2238	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isms 15554	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   X. cxp 4772   |` cres 4776   ` cfv 5377   Basecbs 13352   distcds 13440   TopOpenctopn 13594   Metcmet 14874   *MetSpcxms 15437   MetSpcms 15438

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-ndx 13355  df-slot 13356  df-base 13358  df-tset 13450  df-rest 13595  df-topn 13596  df-top 15099  df-topon 15112  df-topsp 15132  df-xms 15440  df-ms 15441

This theorem is used by: (None)