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Theorem dprdlub 15504
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 2380 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2380 . . . 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 15481 . . 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 643 . 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 2380 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
81adantr 452 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  G dom DProd  S )
9 dprdgrp 15483 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
10 grpmnd 14737 . . . . . 6  |-  ( G  e.  Grp  ->  G  e.  Mnd )
118, 9, 103syl 19 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  G  e.  Mnd )
122adantr 452 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  dom  S  =  I )
13 reldmdprd 15478 . . . . . . . 8  |-  Rel  dom DProd
1413brrelex2i 4852 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
15 dmexg 5063 . . . . . . 7  |-  ( S  e.  _V  ->  dom  S  e.  _V )
168, 14, 153syl 19 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  dom  S  e. 
_V )
1712, 16eqeltrrd 2455 . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  T  e.  (SubGrp `  G ) )
20 subgsubm 14882 . . . . . 6  |-  ( T  e.  (SubGrp `  G
)  ->  T  e.  (SubMnd `  G ) )
2119, 20syl 16 . . . . 5  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  T  e.  (SubMnd `  G ) )
22 simpr 448 . . . . . . . 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 2380 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
244, 8, 12, 22, 23dprdff 15490 . . . . . . 7  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f :
I --> ( Base `  G
) )
25 ffn 5524 . . . . . . 7  |-  ( f : I --> ( Base `  G )  ->  f  Fn  I )
2624, 25syl 16 . . . . . 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 696 . . . . . . . 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 15491 . . . . . . . 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 3285 . . . . . . 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 2725 . . . . . 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 5826 . . . . . 6  |-  ( f : I --> T  <->  ( f  Fn  I  /\  A. k  e.  I  ( f `  k )  e.  T
) )
3326, 31, 32sylanbrc 646 . . . . 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 15493 . . . . 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 15492 . . . . 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 15443 . . . 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 2380 . . . 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 5825 . . 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 5530 . . 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 16 . 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 3318 1  |-  ( ph  ->  ( G DProd  S ) 
C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   _Vcvv 2892    \ cdif 3253    C_ wss 3256   {csn 3750   class class class wbr 4146    e. cmpt 4200   `'ccnv 4810   dom cdm 4811   ran crn 4812   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   X_cixp 6992   Fincfn 7038   Basecbs 13389   0gc0g 13643    gsumg cgsu 13644   Mndcmnd 14604   Grpcgrp 14605  SubMndcsubmnd 14657  SubGrpcsubg 14858  Cntzccntz 15034   DProd cdprd 15474
This theorem is referenced by:  dprdspan  15505  dprdz  15508  dprdcntz2  15516  dprd2dlem1  15519  dprdsplit  15526  ablfac1eu  15551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-0g 13647  df-gsum 13648  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-subg 14861  df-cntz 15036  df-dprd 15476
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