Theorem mspropd 14657

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 14656	. . 3 |- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) )
6	1	sqxpeqd 4686	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
7	6	reseq2d 4943	. . . . . 6 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
8	3, 7	eqtr3d 2228	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	2	sqxpeqd 4686	. . . . . 6 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
10	9	reseq2d 4943	. . . . 5 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
11	8, 10	eqtr3d 2228	. . . 4 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	1, 2	eqtr3d 2228	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
13	12	fveq2d 5559	. . . 4 |- ( ph -> ( Met ` ( Base ` K ) ) = ( Met ` ( Base ` L ) ) )
14	11, 13	eleq12d 2264	. . 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 2193	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2193	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2193	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isms 14632	. 2 |- ( K e. MetSp <-> ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) )
20		eqid 2193	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2193	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2193	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isms 14632	. 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 1364   e. wcel 2164   X. cxp 4658   |` cres 4662   ` cfv 5255   Basecbs 12621   distcds 12707   TopOpenctopn 12854   Metcmet 14036   *MetSpcxms 14515   MetSpcms 14516

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-ndx 12624  df-slot 12625  df-base 12627  df-tset 12717  df-rest 12855  df-topn 12856  df-top 14177  df-topon 14190  df-topsp 14210  df-xms 14518  df-ms 14519

This theorem is referenced by: (None)