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Theorem dpjfval 15290
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 15234 . . . 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 4881 . . . . 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 2315 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  I
)
9 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
g  =  G )
109fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  ( proj
1 `  G )
)
11 dpjfval.q . . . . . 6  |-  Q  =  ( proj 1 `  G )
1210, 11syl6eqr 2333 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  Q )
134fveq1d 5527 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s `  i
)  =  ( S `
 i ) )
148difeq1d 3293 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( dom  s  \  { i } )  =  ( I  \  { i } ) )
154, 14reseq12d 4956 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s  |`  ( dom  s  \  { i } ) )  =  ( S  |`  (
I  \  { i } ) ) )
169, 15oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( g DProd  ( s  |`  ( dom  s  \  { i } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { i } ) ) ) )
1712, 13, 16oveq123d 5879 . . . 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 4098 . . 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 3653 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  { g }  =  { G } )
2120imaeq2d 5012 . . 3  |-  ( (
ph  /\  g  =  G )  ->  ( dom DProd 
" { g } )  =  ( dom DProd  " { G } ) )
22 dpjfval.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
23 dprdgrp 15240 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
2422, 23syl 15 . . 3  |-  ( ph  ->  G  e.  Grp )
25 reldmdprd 15235 . . . . 5  |-  Rel  dom DProd
26 elrelimasn 5037 . . . . 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 4730 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
30 dmexg 4939 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3122, 29, 303syl 18 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
326, 31eqeltrrd 2358 . . . 4  |-  ( ph  ->  I  e.  _V )
33 mptexg 5745 . . . 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 5974 . 2  |-  ( ph  ->  ( GdProj S )  =  ( i  e.  I  |->  ( ( S `
 i ) Q ( G DProd  ( S  |`  ( I  \  {
i } ) ) ) ) ) )
361, 35syl5eq 2327 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   {csn 3640   class class class wbr 4023    e. cmpt 4077   dom cdm 4689    |` cres 4691   "cima 4692   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Grpcgrp 14362   proj
1cpj1 14946   DProd cdprd 15231  dProjcdpj 15232
This theorem is referenced by:  dpjval  15291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-dprd 15233  df-dpj 15234
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