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Theorem dprd2dlem2 15275
Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dprd2d.1  |-  ( ph  ->  Rel  A )
dprd2d.2  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
dprd2d.3  |-  ( ph  ->  dom  A  C_  I
)
dprd2d.4  |-  ( (
ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
dprd2d.5  |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd 
( j  e.  ( A " { i } )  |->  ( i S j ) ) ) ) )
dprd2d.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
dprd2dlem2  |-  ( (
ph  /\  X  e.  A )  ->  ( S `  X )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) ) )
Distinct variable groups:    i, j, A    i, G, j    i, I    i, K    ph, i, j    S, i, j    i, X, j
Allowed substitution hints:    I( j)    K( j)

Proof of Theorem dprd2dlem2
StepHypRef Expression
1 df-ov 5861 . . 3  |-  ( ( 1st `  X ) S ( 2nd `  X
) )  =  ( S `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
2 dprd2d.1 . . . . . . . 8  |-  ( ph  ->  Rel  A )
3 1st2nd 6166 . . . . . . . 8  |-  ( ( Rel  A  /\  X  e.  A )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
42, 3sylan 457 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
5 simpr 447 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  X  e.  A )
64, 5eqeltrrd 2358 . . . . . 6  |-  ( (
ph  /\  X  e.  A )  ->  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  e.  A )
7 df-br 4024 . . . . . 6  |-  ( ( 1st `  X ) A ( 2nd `  X
)  <->  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  e.  A
)
86, 7sylibr 203 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  ( 1st `  X ) A ( 2nd `  X
) )
92adantr 451 . . . . . 6  |-  ( (
ph  /\  X  e.  A )  ->  Rel  A )
10 elrelimasn 5037 . . . . . 6  |-  ( Rel 
A  ->  ( ( 2nd `  X )  e.  ( A " {
( 1st `  X
) } )  <->  ( 1st `  X ) A ( 2nd `  X ) ) )
119, 10syl 15 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  (
( 2nd `  X
)  e.  ( A
" { ( 1st `  X ) } )  <-> 
( 1st `  X
) A ( 2nd `  X ) ) )
128, 11mpbird 223 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( 2nd `  X )  e.  ( A " {
( 1st `  X
) } ) )
13 oveq2 5866 . . . . 5  |-  ( j  =  ( 2nd `  X
)  ->  ( ( 1st `  X ) S j )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
14 eqid 2283 . . . . 5  |-  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )
15 ovex 5883 . . . . 5  |-  ( ( 1st `  X ) S j )  e. 
_V
1613, 14, 15fvmpt3i 5605 . . . 4  |-  ( ( 2nd `  X )  e.  ( A " { ( 1st `  X
) } )  -> 
( ( j  e.  ( A " {
( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) `  ( 2nd `  X ) )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
1712, 16syl 15 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  (
( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) `
 ( 2nd `  X
) )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
184fveq2d 5529 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( S `  X )  =  ( S `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
)
191, 17, 183eqtr4a 2341 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) `
 ( 2nd `  X
) )  =  ( S `  X ) )
20 dprd2d.3 . . . . . 6  |-  ( ph  ->  dom  A  C_  I
)
2120adantr 451 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  dom  A 
C_  I )
22 1stdm 6167 . . . . . 6  |-  ( ( Rel  A  /\  X  e.  A )  ->  ( 1st `  X )  e. 
dom  A )
232, 22sylan 457 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  ( 1st `  X )  e. 
dom  A )
2421, 23sseldd 3181 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( 1st `  X )  e.  I )
25 dprd2d.4 . . . . . 6  |-  ( (
ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) ) )
2625ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. i  e.  I  G dom DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) )
2726adantr 451 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  A. i  e.  I  G dom DProd  ( j  e.  ( A
" { i } )  |->  ( i S j ) ) )
28 sneq 3651 . . . . . . . 8  |-  ( i  =  ( 1st `  X
)  ->  { i }  =  { ( 1st `  X ) } )
2928imaeq2d 5012 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( A " { i } )  =  ( A " { ( 1st `  X
) } ) )
30 oveq1 5865 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( i S j )  =  ( ( 1st `  X
) S j ) )
3129, 30mpteq12dv 4098 . . . . . 6  |-  ( i  =  ( 1st `  X
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) )
3231breq2d 4035 . . . . 5  |-  ( i  =  ( 1st `  X
)  ->  ( G dom DProd  ( j  e.  ( A " { i } )  |->  ( i S j ) )  <-> 
G dom DProd  ( j  e.  ( A " {
( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) ) )
3332rspcv 2880 . . . 4  |-  ( ( 1st `  X )  e.  I  ->  ( A. i  e.  I  G dom DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  ->  G dom DProd  ( j  e.  ( A
" { ( 1st `  X ) } ) 
|->  ( ( 1st `  X
) S j ) ) ) )
3424, 27, 33sylc 56 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  G dom DProd  ( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) )
3515, 14dmmpti 5373 . . . 4  |-  dom  (
j  e.  ( A
" { ( 1st `  X ) } ) 
|->  ( ( 1st `  X
) S j ) )  =  ( A
" { ( 1st `  X ) } )
3635a1i 10 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  dom  ( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) )  =  ( A " { ( 1st `  X
) } ) )
3734, 36, 12dprdub 15260 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) `
 ( 2nd `  X
) )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) ) )
3819, 37eqsstr3d 3213 1  |-  ( (
ph  /\  X  e.  A )  ->  ( S `  X )  C_  ( G DProd  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   "cima 4692   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  mrClscmrc 13485  SubGrpcsubg 14615   DProd cdprd 15231
This theorem is referenced by:  dprd2dlem1  15276  dprd2da  15277
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  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-mulg 14492  df-subg 14618  df-cntz 14793  df-cmn 15091  df-dprd 15233
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