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Theorem dprdlub 15277
Description: The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdlub.1  |-  ( ph  ->  G dom DProd  S )
dprdlub.2  |-  ( ph  ->  dom  S  =  I )
dprdlub.3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
dprdlub.4  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  T )
Assertion
Ref Expression
dprdlub  |-  ( ph  ->  ( G DProd  S ) 
C_  T )
Distinct variable groups:    k, G    k, I    ph, k    S, k    T, k

Proof of Theorem dprdlub
Dummy variables  f  h  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdlub.1 . . 3  |-  ( ph  ->  G dom DProd  S )
2 dprdlub.2 . . 3  |-  ( ph  ->  dom  S  =  I )
3 eqid 2296 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2296 . . . 4  |-  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  =  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }
53, 4dprdval 15254 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) ) )
61, 2, 5syl2anc 642 . 2  |-  ( ph  ->  ( G DProd  S )  =  ran  ( f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) ) )
7 eqid 2296 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
81adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  G dom DProd  S )
9 dprdgrp 15256 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
10 grpmnd 14510 . . . . . 6  |-  ( G  e.  Grp  ->  G  e.  Mnd )
118, 9, 103syl 18 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  G  e.  Mnd )
122adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  dom  S  =  I )
13 reldmdprd 15251 . . . . . . . 8  |-  Rel  dom DProd
1413brrelex2i 4746 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
15 dmexg 4955 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
168, 14, 153syl 18 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  dom  S  e. 
_V )
1712, 16eqeltrrd 2371 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  I  e.  _V )
18 dprdlub.3 . . . . . . 7  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
1918adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  T  e.  (SubGrp `  G ) )
20 subgsubm 14655 . . . . . 6  |-  ( T  e.  (SubGrp `  G
)  ->  T  e.  (SubMnd `  G ) )
2119, 20syl 15 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  T  e.  (SubMnd `  G ) )
22 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )
23 eqid 2296 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
244, 8, 12, 22, 23dprdff 15263 . . . . . . 7  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f :
I --> ( Base `  G
) )
25 ffn 5405 . . . . . . 7  |-  ( f : I --> ( Base `  G )  ->  f  Fn  I )
2624, 25syl 15 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f  Fn  I )
27 dprdlub.4 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  T )
2827adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } )  /\  k  e.  I
)  ->  ( S `  k )  C_  T
)
294, 8, 12, 22dprdfcl 15264 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } )  /\  k  e.  I
)  ->  ( f `  k )  e.  ( S `  k ) )
3028, 29sseldd 3194 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } )  /\  k  e.  I
)  ->  ( f `  k )  e.  T
)
3130ralrimiva 2639 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  A. k  e.  I  ( f `  k )  e.  T
)
32 ffnfv 5701 . . . . . 6  |-  ( f : I --> T  <->  ( f  Fn  I  /\  A. k  e.  I  ( f `  k )  e.  T
) )
3326, 31, 32sylanbrc 645 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f :
I --> T )
344, 8, 12, 22, 7dprdfcntz 15266 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  ran  f  C_  ( (Cntz `  G ) `  ran  f ) )
354, 8, 12, 22dprdffi 15265 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  ( `' f " ( _V  \  { ( 0g `  G ) } ) )  e.  Fin )
363, 7, 11, 17, 21, 33, 34, 35gsumzsubmcl 15216 . . . 4  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  ( G  gsumg  f )  e.  T )
37 eqid 2296 . . . 4  |-  ( f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) )  =  ( f  e.  {
h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) )
3836, 37fmptd 5700 . . 3  |-  ( ph  ->  ( f  e.  {
h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) ) : { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } --> T )
39 frn 5411 . . 3  |-  ( ( f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) ) : { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } --> T  ->  ran  ( f  e.  {
h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
|->  ( G  gsumg  f ) )  C_  T )
4038, 39syl 15 . 2  |-  ( ph  ->  ran  ( f  e. 
{ h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin }  |->  ( G  gsumg  f ) )  C_  T )
416, 40eqsstrd 3225 1  |-  ( ph  ->  ( G DProd  S ) 
C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Fincfn 6879   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378  SubMndcsubmnd 14430  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdspan  15278  dprdz  15281  dprdcntz2  15289  dprd2dlem1  15292  dprdsplit  15299  ablfac1eu  15324
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-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-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-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-cntz 14809  df-dprd 15249
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