Theorem ovresd 6220

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 6219	. 2 |- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   X. cxp 4767   |` cres 4771  (class class class)co 6075

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-res 4781  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  psmetres2  15357  xmetres2  15403  xmssym  15493  xmstri2  15494  mstri2  15495  xmstri  15496  mstri  15497  xmstri3  15498  mstri3  15499  msrtri  15500  limcimolemlt  15688  cnplimcim  15691  cnplimclemr  15693  limccnpcntop  15699  limccnp2lem  15700