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Theorem dprd2dlem2 15291
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 5877 . . 3  |-  ( ( 1st `  X ) S ( 2nd `  X
) )  =  ( S `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
2 dprd2d.1 . . . . . . . 8  |-  ( ph  ->  Rel  A )
3 1st2nd 6182 . . . . . . . 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 2371 . . . . . 6  |-  ( (
ph  /\  X  e.  A )  ->  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  e.  A )
7 df-br 4040 . . . . . 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 5053 . . . . . 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 5882 . . . . 5  |-  ( j  =  ( 2nd `  X
)  ->  ( ( 1st `  X ) S j )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
14 eqid 2296 . . . . 5  |-  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )
15 ovex 5899 . . . . 5  |-  ( ( 1st `  X ) S j )  e. 
_V
1613, 14, 15fvmpt3i 5621 . . . 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 5545 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( S `  X )  =  ( S `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
)
191, 17, 183eqtr4a 2354 . 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 6183 . . . . . 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 3194 . . . 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 2639 . . . . 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 3664 . . . . . . . 8  |-  ( i  =  ( 1st `  X
)  ->  { i }  =  { ( 1st `  X ) } )
2928imaeq2d 5028 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( A " { i } )  =  ( A " { ( 1st `  X
) } ) )
30 oveq1 5881 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( i S j )  =  ( ( 1st `  X
) S j ) )
3129, 30mpteq12dv 4114 . . . . . 6  |-  ( i  =  ( 1st `  X
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) )
3231breq2d 4051 . . . . 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 2893 . . . 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 5389 . . . 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 15276 . 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 3226 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   "cima 4708   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  mrClscmrc 13501  SubGrpcsubg 14631   DProd cdprd 15247
This theorem is referenced by:  dprd2dlem1  15292  dprd2da  15293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-mulg 14508  df-subg 14634  df-cntz 14809  df-cmn 15107  df-dprd 15249
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