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Theorem dpjfval 15339
Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dpjfval.1  |-  ( ph  ->  G dom DProd  S )
dpjfval.2  |-  ( ph  ->  dom  S  =  I )
dpjfval.p  |-  P  =  ( GdProj S )
dpjfval.q  |-  Q  =  ( proj 1 `  G )
Assertion
Ref Expression
dpjfval  |-  ( ph  ->  P  =  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I  \  { i } ) ) ) ) ) )
Distinct variable groups:    i, G    ph, i    i, I    S, i
Allowed substitution hints:    P( i)    Q( i)

Proof of Theorem dpjfval
Dummy variables  g 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dpjfval.p . 2  |-  P  =  ( GdProj S )
2 df-dpj 15283 . . . 4  |- dProj  =  ( g  e.  Grp , 
s  e.  ( dom DProd  " { g } ) 
|->  ( i  e.  dom  s  |->  ( ( s `
 i ) (
proj 1 `  g ) ( g DProd  ( s  |`  ( dom  s  \  { i } ) ) ) ) ) )
32a1i 10 . . 3  |-  ( ph  -> dProj  =  ( g  e. 
Grp ,  s  e.  ( dom DProd  " { g } )  |->  ( i  e. 
dom  s  |->  ( ( s `  i ) ( proj 1 `  g ) ( g DProd 
( s  |`  ( dom  s  \  { i } ) ) ) ) ) ) )
4 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
s  =  S )
54dmeqd 4918 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  dom  S )
6 dpjfval.2 . . . . . 6  |-  ( ph  ->  dom  S  =  I )
76adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  S  =  I )
85, 7eqtrd 2348 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  I
)
9 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
g  =  G )
109fveq2d 5567 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  ( proj
1 `  G )
)
11 dpjfval.q . . . . . 6  |-  Q  =  ( proj 1 `  G )
1210, 11syl6eqr 2366 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  Q )
134fveq1d 5565 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s `  i
)  =  ( S `
 i ) )
148difeq1d 3327 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( dom  s  \  { i } )  =  ( I  \  { i } ) )
154, 14reseq12d 4993 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s  |`  ( dom  s  \  { i } ) )  =  ( S  |`  (
I  \  { i } ) ) )
169, 15oveq12d 5918 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( g DProd  ( s  |`  ( dom  s  \  { i } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { i } ) ) ) )
1712, 13, 16oveq123d 5921 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( ( s `  i ) ( proj
1 `  g )
( g DProd  ( s  |`  ( dom  s  \  { i } ) ) ) )  =  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I  \  { i } ) ) ) ) )
188, 17mpteq12dv 4135 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( i  e.  dom  s  |->  ( ( s `
 i ) (
proj 1 `  g ) ( g DProd  ( s  |`  ( dom  s  \  { i } ) ) ) ) )  =  ( i  e.  I  |->  ( ( S `
 i ) Q ( G DProd  ( S  |`  ( I  \  {
i } ) ) ) ) ) )
19 simpr 447 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
2019sneqd 3687 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  { g }  =  { G } )
2120imaeq2d 5049 . . 3  |-  ( (
ph  /\  g  =  G )  ->  ( dom DProd 
" { g } )  =  ( dom DProd  " { G } ) )
22 dpjfval.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
23 dprdgrp 15289 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
2422, 23syl 15 . . 3  |-  ( ph  ->  G  e.  Grp )
25 reldmdprd 15284 . . . . 5  |-  Rel  dom DProd
26 elrelimasn 5074 . . . . 5  |-  ( Rel 
dom DProd  ->  ( S  e.  ( dom DProd  " { G } )  <->  G dom DProd  S ) )
2725, 26ax-mp 8 . . . 4  |-  ( S  e.  ( dom DProd  " { G } )  <->  G dom DProd  S )
2822, 27sylibr 203 . . 3  |-  ( ph  ->  S  e.  ( dom DProd  " { G } ) )
2925brrelex2i 4767 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
30 dmexg 4976 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3122, 29, 303syl 18 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
326, 31eqeltrrd 2391 . . . 4  |-  ( ph  ->  I  e.  _V )
33 mptexg 5786 . . . 4  |-  ( I  e.  _V  ->  (
i  e.  I  |->  ( ( S `  i
) Q ( G DProd 
( S  |`  (
I  \  { i } ) ) ) ) )  e.  _V )
3432, 33syl 15 . . 3  |-  ( ph  ->  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I  \  { i } ) ) ) ) )  e.  _V )
353, 18, 21, 24, 28, 34ovmpt2dx 6016 . 2  |-  ( ph  ->  ( GdProj S )  =  ( i  e.  I  |->  ( ( S `
 i ) Q ( G DProd  ( S  |`  ( I  \  {
i } ) ) ) ) ) )
361, 35syl5eq 2360 1  |-  ( ph  ->  P  =  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I  \  { i } ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183   {csn 3674   class class class wbr 4060    e. cmpt 4114   dom cdm 4726    |` cres 4728   "cima 4729   Rel wrel 4731   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   Grpcgrp 14411   proj
1cpj1 14995   DProd cdprd 15280  dProjcdpj 15281
This theorem is referenced by:  dpjval  15340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-ixp 6861  df-dprd 15282  df-dpj 15283
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