Theorem mspropd 15502

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 15501	. . 3 |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )
6	1	sqxpeqd 4795	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
7	6	reseq2d 5058	. . . . . 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 4795	. . . . . 6 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
10	9	reseq2d 5058	. . . . 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 5694	. . . 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 15477	. 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 15477	. 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 1402   e. wcel 2209   X. cxp 4767   |` cres 4771   ` cfv 5372   Basecbs 13330   distcds 13417   TopOpenctopn 13571   Metcmet 14846   *MetSpcxms 15360   MetSpcms 15361

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-ndx 13333  df-slot 13334  df-base 13336  df-tset 13427  df-rest 13572  df-topn 13573  df-top 15022  df-topon 15035  df-topsp 15055  df-xms 15363  df-ms 15364

This theorem is referenced by: (None)