Theorem ovresd 5661

Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
ovresd.1	|- ( ph -> A e. X )
ovresd.2	|- ( ph -> B e. X )

Assertion

Ref	Expression
ovresd	|- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )

Proof of Theorem ovresd

Step	Hyp	Ref	Expression
1		ovresd.1	. 2 |- ( ph -> A e. X )
2		ovresd.2	. 2 |- ( ph -> B e. X )
3		ovres 5660	. 2 |- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4	1, 2, 3	syl2anc 403	1 |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   X. cxp 4361   |` cres 4365  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-res 4375  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by: (None)