Theorem xmspropd 14982

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 2240	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4		xmspropd.4	. . . 4 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
5	3, 4	tpspropd 14541	. . 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 4702	. . . . . . . 8 |- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) )
8	7	reseq2d 4960	. . . . . . 7 |- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
9	6, 8	eqtr3d 2240	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) )
10	2	sqxpeqd 4702	. . . . . . 7 |- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) )
11	10	reseq2d 4960	. . . . . 6 |- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
12	9, 11	eqtr3d 2240	. . . . 5 |- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) )
13	12	fveq2d 5582	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) = ( MetOpen ` ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) )
14	4, 13	eqeq12d 2220	. . 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 2205	. . 3 |- ( TopOpen ` K ) = ( TopOpen ` K )
17		eqid 2205	. . 3 |- ( Base ` K ) = ( Base ` K )
18		eqid 2205	. . 3 |- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) )
19	16, 17, 18	isxms 14956	. 2 |- ( K e. *MetSp <-> ( K e. TopSp /\ ( TopOpen ` K ) = ( MetOpen ` ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) ) )
20		eqid 2205	. . 3 |- ( TopOpen ` L ) = ( TopOpen ` L )
21		eqid 2205	. . 3 |- ( Base ` L ) = ( Base ` L )
22		eqid 2205	. . 3 |- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) )
23	20, 21, 22	isxms 14956	. 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 2176   X. cxp 4674   |` cres 4678   ` cfv 5272   Basecbs 12865   distcds 12951   TopOpenctopn 13105   MetOpencmopn 14336   TopSpctps 14535   *MetSpcxms 14841

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-9 9104  df-ndx 12868  df-slot 12869  df-base 12871  df-tset 12961  df-rest 13106  df-topn 13107  df-top 14503  df-topon 14516  df-topsp 14536  df-xms 14844

This theorem is referenced by:  mspropd  14983