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Theorem dpjfval 15615
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 15559 . . . 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 735 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
s  =  S )
54dmeqd 5074 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  dom  S )
6 dpjfval.2 . . . . . 6  |-  ( ph  ->  dom  S  =  I )
76adantr 453 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  S  =  I )
85, 7eqtrd 2470 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  ->  dom  s  =  I
)
9 simprl 734 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
g  =  G )
109fveq2d 5734 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  ( proj
1 `  G )
)
11 dpjfval.q . . . . . 6  |-  Q  =  ( proj 1 `  G )
1210, 11syl6eqr 2488 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( proj 1 `  g
)  =  Q )
134fveq1d 5732 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s `  i
)  =  ( S `
 i ) )
148difeq1d 3466 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( dom  s  \  { i } )  =  ( I  \  { i } ) )
154, 14reseq12d 5149 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( s  |`  ( dom  s  \  { i } ) )  =  ( S  |`  (
I  \  { i } ) ) )
169, 15oveq12d 6101 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  s  =  S ) )  -> 
( g DProd  ( s  |`  ( dom  s  \  { i } ) ) )  =  ( G DProd  ( S  |`  ( I  \  { i } ) ) ) )
1712, 13, 16oveq123d 6104 . . . 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 4289 . . 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 449 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
2019sneqd 3829 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  { g }  =  { G } )
2120imaeq2d 5205 . . 3  |-  ( (
ph  /\  g  =  G )  ->  ( dom DProd 
" { g } )  =  ( dom DProd  " { G } ) )
22 dpjfval.1 . . . 4  |-  ( ph  ->  G dom DProd  S )
23 dprdgrp 15565 . . . 4  |-  ( G dom DProd  S  ->  G  e. 
Grp )
2422, 23syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
25 reldmdprd 15560 . . . . 5  |-  Rel  dom DProd
26 elrelimasn 5230 . . . . 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 205 . . 3  |-  ( ph  ->  S  e.  ( dom DProd  " { G } ) )
2925brrelex2i 4921 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
30 dmexg 5132 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3122, 29, 303syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
326, 31eqeltrrd 2513 . . . 4  |-  ( ph  ->  I  e.  _V )
33 mptexg 5967 . . . 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 6202 . 2  |-  ( ph  ->  ( GdProj S )  =  ( i  e.  I  |->  ( ( S `
 i ) Q ( G DProd  ( S  |`  ( I  \  {
i } ) ) ) ) ) )
361, 35syl5eq 2482 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   {csn 3816   class class class wbr 4214    e. cmpt 4268   dom cdm 4880    |` cres 4882   "cima 4883   Rel wrel 4885   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   Grpcgrp 14687   proj
1cpj1 15271   DProd cdprd 15556  dProjcdpj 15557
This theorem is referenced by:  dpjval  15616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-ixp 7066  df-dprd 15558  df-dpj 15559
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