Theorem ovresd 6801

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 6800	. 2 |- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4	1, 2, 3	syl2anc 693	1 |- ( ph -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116  (class class class)co 6650

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  sscres  16483  fullsubc  16510  fullresc  16511  funcres2c  16561  psmetres2  22119  xmetres2  22166  prdsdsf  22172  xpsdsval  22186  xmssym  22270  xmstri2  22271  mstri2  22272  xmstri  22273  mstri  22274  xmstri3  22275  mstri3  22276  msrtri  22277  tmsxpsval  22343  ngptgp  22440  nlmvscn  22491  nrginvrcn  22496  nghmcn  22549  cnmpt1ds  22645  cnmpt2ds  22646  ipcn  23045  caussi  23095  causs  23096  minveclem2  23197  minveclem3b  23199  minveclem3  23200  minveclem4  23203  minveclem6  23205  ftc1lem6  23804  ulmdvlem1  24154  abelth  24195  cxpcn3  24489  rlimcnp  24692  hhssnv  28121  madjusmdetlem3  29895  qqhcn  30035  qqhucn  30036  ftc1cnnc  33484  ismtyres  33607  isdrngo2  33757  rngchom  41967  ringchom  42013  irinitoringc  42069  rhmsubclem4  42089