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Theorem dpjfval 15568
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 15512 . . . 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 11 . . 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 734 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
s  =  S )
54dmeqd 5031 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  dom  S )
6 dpjfval.2 . . . . . 6  |-  ( ph  ->  dom  S  =  I )
76adantr 452 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  S  =  I )
85, 7eqtrd 2436 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  I
)
9 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
g  =  G )
109fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  ( proj
1 `  G )
)
11 dpjfval.q . . . . . 6  |-  Q  =  ( proj 1 `  G )
1210, 11syl6eqr 2454 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  Q )
134fveq1d 5689 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s `  i
)  =  ( S `
 i ) )
148difeq1d 3424 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( dom  s  \  { i } )  =  ( I  \  { i } ) )
154, 14reseq12d 5106 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s  |`  ( dom  s  \  { i } ) )  =  ( S  |`  (
I  \  { i } ) ) )
169, 15oveq12d 6058 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( g DProd  ( s  |`  ( dom  s  \  { i } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { i } ) ) ) )
1712, 13, 16oveq123d 6061 . . . 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 4247 . . 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 448 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
2019sneqd 3787 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  { g }  =  { G } )
2120imaeq2d 5162 . . 3  |-  ( (
ph  /\  g  =  G )  ->  ( dom DProd 
" { g } )  =  ( dom DProd  " { G } ) )
22 dpjfval.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
23 dprdgrp 15518 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
2422, 23syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
25 reldmdprd 15513 . . . . 5  |-  Rel  dom DProd
26 elrelimasn 5187 . . . . 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 204 . . 3  |-  ( ph  ->  S  e.  ( dom DProd  " { G } ) )
2925brrelex2i 4878 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
30 dmexg 5089 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3122, 29, 303syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
326, 31eqeltrrd 2479 . . . 4  |-  ( ph  ->  I  e.  _V )
33 mptexg 5924 . . . 4  |-  ( I  e.  _V  ->  (
i  e.  I  |->  ( ( S `  i
) Q ( G DProd 
( S  |`  (
I  \  { i } ) ) ) ) )  e.  _V )
3432, 33syl 16 . . 3  |-  ( ph  ->  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I  \  { i } ) ) ) ) )  e.  _V )
353, 18, 21, 24, 28, 34ovmpt2dx 6159 . 2  |-  ( ph  ->  ( GdProj S )  =  ( i  e.  I  |->  ( ( S `
 i ) Q ( G DProd  ( S  |`  ( I  \  {
i } ) ) ) ) ) )
361, 35syl5eq 2448 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277   {csn 3774   class class class wbr 4172    e. cmpt 4226   dom cdm 4837    |` cres 4839   "cima 4840   Rel wrel 4842   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   Grpcgrp 14640   proj
1cpj1 15224   DProd cdprd 15509  dProjcdpj 15510
This theorem is referenced by:  dpjval  15569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-ixp 7023  df-dprd 15511  df-dpj 15512
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