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

Theorem mhmima 14440
Description: The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
mhmima  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  e.  (SubMnd `  N ) )

Proof of Theorem mhmima
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 5025 . . 3  |-  ( F
" X )  C_  ran  F
2 eqid 2283 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2283 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
42, 3mhmf 14420 . . . . 5  |-  ( F  e.  ( M MndHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
54adantr 451 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
6 frn 5395 . . . 4  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  ran  F  C_  ( Base `  N )
)
75, 6syl 15 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ran  F  C_  ( Base `  N )
)
81, 7syl5ss 3190 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  C_  ( Base `  N ) )
9 eqid 2283 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 eqid 2283 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
119, 10mhm0 14423 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
1211adantr 451 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
13 ffn 5389 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
145, 13syl 15 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F  Fn  ( Base `  M )
)
152submss 14427 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  X  C_  ( Base `  M ) )
1615adantl 452 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  X  C_  ( Base `  M ) )
179subm0cl 14429 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  ( 0g `  M )  e.  X
)
1817adantl 452 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  M )  e.  X
)
19 fnfvima 5756 . . . 4  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  X
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2014, 16, 18, 19syl3anc 1182 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2112, 20eqeltrrd 2358 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  N )  e.  ( F " X ) )
22 simpll 730 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  e.  ( M MndHom  N ) )
2316adantr 451 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  X  C_  ( Base `  M ) )
24 simprl 732 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  X )
2523, 24sseldd 3181 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  ( Base `  M )
)
26 simprr 733 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  X )
2723, 26sseldd 3181 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  ( Base `  M )
)
28 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
29 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  N )
302, 28, 29mhmlin 14422 . . . . . . . . 9  |-  ( ( F  e.  ( M MndHom  N )  /\  z  e.  ( Base `  M
)  /\  x  e.  ( Base `  M )
)  ->  ( F `  ( z ( +g  `  M ) x ) )  =  ( ( F `  z ) ( +g  `  N
) ( F `  x ) ) )
3122, 25, 27, 30syl3anc 1182 . . . . . . . 8  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( F `  ( z ( +g  `  M ) x ) )  =  ( ( F `  z ) ( +g  `  N
) ( F `  x ) ) )
3214adantr 451 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  Fn  ( Base `  M )
)
3328submcl 14430 . . . . . . . . . . 11  |-  ( ( X  e.  (SubMnd `  M )  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( +g  `  M
) x )  e.  X )
34333expb 1152 . . . . . . . . . 10  |-  ( ( X  e.  (SubMnd `  M )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
3534adantll 694 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
36 fnfvima 5756 . . . . . . . . 9  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
)  /\  ( z
( +g  `  M ) x )  e.  X
)  ->  ( F `  ( z ( +g  `  M ) x ) )  e.  ( F
" X ) )
3732, 23, 35, 36syl3anc 1182 . . . . . . . 8  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( F `  ( z ( +g  `  M ) x ) )  e.  ( F
" X ) )
3831, 37eqeltrrd 2358 . . . . . . 7  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) )
3938anassrs 629 . . . . . 6  |-  ( ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M ) )  /\  z  e.  X )  /\  x  e.  X
)  ->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) )
4039ralrimiva 2626 . . . . 5  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  z  e.  X )  ->  A. x  e.  X  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) )
41 oveq2 5866 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) ) )
4241eleq1d 2349 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) ) )
4342ralima 5758 . . . . . . 7  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
) )  ->  ( A. y  e.  ( F " X ) ( ( F `  z
) ( +g  `  N
) y )  e.  ( F " X
)  <->  A. x  e.  X  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) )  e.  ( F " X ) ) )
4414, 16, 43syl2anc 642 . . . . . 6  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( A. y  e.  ( F " X ) ( ( F `  z ) ( +g  `  N
) y )  e.  ( F " X
)  <->  A. x  e.  X  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) )  e.  ( F " X ) ) )
4544adantr 451 . . . . 5  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  z  e.  X )  ->  ( A. y  e.  ( F " X ) ( ( F `  z
) ( +g  `  N
) y )  e.  ( F " X
)  <->  A. x  e.  X  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) )  e.  ( F " X ) ) )
4640, 45mpbird 223 . . . 4  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  z  e.  X )  ->  A. y  e.  ( F " X
) ( ( F `
 z ) ( +g  `  N ) y )  e.  ( F " X ) )
4746ralrimiva 2626 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  A. z  e.  X  A. y  e.  ( F " X
) ( ( F `
 z ) ( +g  `  N ) y )  e.  ( F " X ) )
48 oveq1 5865 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) y ) )
4948eleq1d 2349 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) y )  e.  ( F " X ) ) )
5049ralbidv 2563 . . . . 5  |-  ( x  =  ( F `  z )  ->  ( A. y  e.  ( F " X ) ( x ( +g  `  N
) y )  e.  ( F " X
)  <->  A. y  e.  ( F " X ) ( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X ) ) )
5150ralima 5758 . . . 4  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
) )  ->  ( A. x  e.  ( F " X ) A. y  e.  ( F " X ) ( x ( +g  `  N
) y )  e.  ( F " X
)  <->  A. z  e.  X  A. y  e.  ( F " X ) ( ( F `  z
) ( +g  `  N
) y )  e.  ( F " X
) ) )
5214, 16, 51syl2anc 642 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( A. x  e.  ( F " X ) A. y  e.  ( F " X
) ( x ( +g  `  N ) y )  e.  ( F " X )  <->  A. z  e.  X  A. y  e.  ( F " X ) ( ( F `  z
) ( +g  `  N
) y )  e.  ( F " X
) ) )
5347, 52mpbird 223 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  A. x  e.  ( F " X
) A. y  e.  ( F " X
) ( x ( +g  `  N ) y )  e.  ( F " X ) )
54 mhmrcl2 14419 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5554adantr 451 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  N  e.  Mnd )
563, 10, 29issubm 14425 . . 3  |-  ( N  e.  Mnd  ->  (
( F " X
)  e.  (SubMnd `  N )  <->  ( ( F " X )  C_  ( Base `  N )  /\  ( 0g `  N
)  e.  ( F
" X )  /\  A. x  e.  ( F
" X ) A. y  e.  ( F " X ) ( x ( +g  `  N
) y )  e.  ( F " X
) ) ) )
5755, 56syl 15 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( ( F " X )  e.  (SubMnd `  N )  <->  ( ( F " X
)  C_  ( Base `  N )  /\  ( 0g `  N )  e.  ( F " X
)  /\  A. x  e.  ( F " X
) A. y  e.  ( F " X
) ( x ( +g  `  N ) y )  e.  ( F " X ) ) ) )
588, 21, 53, 57mpbir3and 1135 1  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  e.  (SubMnd `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   MndHom cmhm 14413  SubMndcsubmnd 14414
This theorem is referenced by:  rhmima  15576
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-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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-submnd 14416
  Copyright terms: Public domain W3C validator