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Theorem dprd2dlem2 15600
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 6086 . . 3  |-  ( ( 1st `  X ) S ( 2nd `  X
) )  =  ( S `  <. ( 1st `  X ) ,  ( 2nd `  X
) >. )
2 dprd2d.1 . . . . . . . 8  |-  ( ph  ->  Rel  A )
3 1st2nd 6395 . . . . . . . 8  |-  ( ( Rel  A  /\  X  e.  A )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
42, 3sylan 459 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  X  =  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
5 simpr 449 . . . . . . 7  |-  ( (
ph  /\  X  e.  A )  ->  X  e.  A )
64, 5eqeltrrd 2513 . . . . . 6  |-  ( (
ph  /\  X  e.  A )  ->  <. ( 1st `  X ) ,  ( 2nd `  X
) >.  e.  A )
7 df-br 4215 . . . . . 6  |-  ( ( 1st `  X ) A ( 2nd `  X
)  <->  <. ( 1st `  X
) ,  ( 2nd `  X ) >.  e.  A
)
86, 7sylibr 205 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  ( 1st `  X ) A ( 2nd `  X
) )
92adantr 453 . . . . . 6  |-  ( (
ph  /\  X  e.  A )  ->  Rel  A )
10 elrelimasn 5230 . . . . . 6  |-  ( Rel 
A  ->  ( ( 2nd `  X )  e.  ( A " {
( 1st `  X
) } )  <->  ( 1st `  X ) A ( 2nd `  X ) ) )
119, 10syl 16 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  (
( 2nd `  X
)  e.  ( A
" { ( 1st `  X ) } )  <-> 
( 1st `  X
) A ( 2nd `  X ) ) )
128, 11mpbird 225 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( 2nd `  X )  e.  ( A " {
( 1st `  X
) } ) )
13 oveq2 6091 . . . . 5  |-  ( j  =  ( 2nd `  X
)  ->  ( ( 1st `  X ) S j )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
14 eqid 2438 . . . . 5  |-  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) )
15 ovex 6108 . . . . 5  |-  ( ( 1st `  X ) S j )  e. 
_V
1613, 14, 15fvmpt3i 5811 . . . 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 16 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  (
( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) `
 ( 2nd `  X
) )  =  ( ( 1st `  X
) S ( 2nd `  X ) ) )
184fveq2d 5734 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( S `  X )  =  ( S `  <. ( 1st `  X
) ,  ( 2nd `  X ) >. )
)
191, 17, 183eqtr4a 2496 . 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 453 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  dom  A 
C_  I )
22 1stdm 6396 . . . . . 6  |-  ( ( Rel  A  /\  X  e.  A )  ->  ( 1st `  X )  e. 
dom  A )
232, 22sylan 459 . . . . 5  |-  ( (
ph  /\  X  e.  A )  ->  ( 1st `  X )  e. 
dom  A )
2421, 23sseldd 3351 . . . 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 2791 . . . . 5  |-  ( ph  ->  A. i  e.  I  G dom DProd  ( j  e.  ( A " {
i } )  |->  ( i S j ) ) )
2726adantr 453 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  A. i  e.  I  G dom DProd  ( j  e.  ( A
" { i } )  |->  ( i S j ) ) )
28 sneq 3827 . . . . . . . 8  |-  ( i  =  ( 1st `  X
)  ->  { i }  =  { ( 1st `  X ) } )
2928imaeq2d 5205 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( A " { i } )  =  ( A " { ( 1st `  X
) } ) )
30 oveq1 6090 . . . . . . 7  |-  ( i  =  ( 1st `  X
)  ->  ( i S j )  =  ( ( 1st `  X
) S j ) )
3129, 30mpteq12dv 4289 . . . . . 6  |-  ( i  =  ( 1st `  X
)  ->  ( j  e.  ( A " {
i } )  |->  ( i S j ) )  =  ( j  e.  ( A " { ( 1st `  X
) } )  |->  ( ( 1st `  X
) S j ) ) )
3231breq2d 4226 . . . . 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 3050 . . . 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 59 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  G dom DProd  ( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) )
3515, 14dmmpti 5576 . . . 4  |-  dom  (
j  e.  ( A
" { ( 1st `  X ) } ) 
|->  ( ( 1st `  X
) S j ) )  =  ( A
" { ( 1st `  X ) } )
3635a1i 11 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  dom  ( j  e.  ( A " { ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) )  =  ( A " { ( 1st `  X
) } ) )
3734, 36, 12dprdub 15585 . 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 3385 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   {csn 3816   <.cop 3819   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   "cima 4883   Rel wrel 4885   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  mrClscmrc 13810  SubGrpcsubg 14940   DProd cdprd 15556
This theorem is referenced by:  dprd2dlem1  15601  dprd2da  15602
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  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  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-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-gsum 13730  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-grp 14814  df-mulg 14817  df-subg 14943  df-cntz 15118  df-cmn 15416  df-dprd 15558
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