Theorem ovresd 6110

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 6109	. 2 |- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   X. cxp 4691   |` cres 4695  (class class class)co 5967

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-res 4705  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  psmetres2  14920  xmetres2  14966  xmssym  15056  xmstri2  15057  mstri2  15058  xmstri  15059  mstri  15060  xmstri3  15061  mstri3  15062  msrtri  15063  limcimolemlt  15251  cnplimcim  15254  cnplimclemr  15256  limccnpcntop  15262  limccnp2lem  15263