Theorem metidv 29935

Description: A and B identify by the metric D if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metidv	|- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A (~Met ` D ) B <-> ( A D B ) = 0 ) )

Proof of Theorem metidv

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . 6 |- ( a = A -> ( a e. X <-> A e. X ) )
2		eleq1 2689	. . . . . 6 |- ( b = B -> ( b e. X <-> B e. X ) )
3	1, 2	bi2anan9 917	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ( a e. X /\ b e. X ) <-> ( A e. X /\ B e. X ) ) )
4		oveq12 6659	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( a D b ) = ( A D B ) )
5	4	eqeq1d 2624	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ( a D b ) = 0 <-> ( A D B ) = 0 ) )
6	3, 5	anbi12d 747	. . . 4 |- ( ( a = A /\ b = B ) -> ( ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) <-> ( ( A e. X /\ B e. X ) /\ ( A D B ) = 0 ) ) )
7		eqid 2622	. . . 4 |- { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } = { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) }
8	6, 7	brabga 4989	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } B <-> ( ( A e. X /\ B e. X ) /\ ( A D B ) = 0 ) ) )
9	8	adantl 482	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } B <-> ( ( A e. X /\ B e. X ) /\ ( A D B ) = 0 ) ) )
10		metidval 29933	. . . 4 |- ( D e. (PsMet ` X ) -> (~Met ` D ) = { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } )
11	10	adantr 481	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> (~Met ` D ) = { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } )
12	11	breqd 4664	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A (~Met ` D ) B <-> A { <. a , b >. | ( ( a e. X /\ b e. X ) /\ ( a D b ) = 0 ) } B ) )
13		ibar 525	. . 3 |- ( ( A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> ( ( A e. X /\ B e. X ) /\ ( A D B ) = 0 ) ) )
14	13	adantl 482	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( ( A D B ) = 0 <-> ( ( A e. X /\ B e. X ) /\ ( A D B ) = 0 ) ) )
15	9, 12, 14	3bitr4d 300	1 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A (~Met ` D ) B <-> ( A D B ) = 0 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   0cc0 9936  PsMetcpsmet 19730  ~Metcmetid 29929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-psmet 19738  df-metid 29931

This theorem is referenced by:  metideq  29936  metider  29937  pstmfval  29939  pstmxmet  29940