MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdlub Unicode version

Theorem dprdlub 15261
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 2283 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2283 . . . 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 15238 . . 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 2283 . . . . 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 15240 . . . . . 6  |-  ( G dom DProd  S  ->  G  e. 
Grp )
10 grpmnd 14494 . . . . . 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 15235 . . . . . . . 8  |-  Rel  dom DProd
1413brrelex2i 4730 . . . . . . 7  |-  ( G dom DProd  S  ->  S  e. 
_V )
15 dmexg 4939 . . . . . . 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 2358 . . . . 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 14639 . . . . . 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 2283 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
244, 8, 12, 22, 23dprdff 15247 . . . . . . 7  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )  ->  f :
I --> ( Base `  G
) )
25 ffn 5389 . . . . . . 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 15248 . . . . . . . 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 3181 . . . . . . 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 2626 . . . . . 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 5685 . . . . . 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 15250 . . . . 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 15249 . . . . 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 15200 . . . 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 2283 . . . 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 5684 . . 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 5395 . . 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 3212 1  |-  ( ph  ->  ( G DProd  S ) 
C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362  SubMndcsubmnd 14414  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdspan  15262  dprdz  15265  dprdcntz2  15273  dprd2dlem1  15276  dprdsplit  15283  ablfac1eu  15308
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-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-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-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-dprd 15233
  Copyright terms: Public domain W3C validator