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Theorem mhmima 14456
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 5041 . . 3  |-  ( F
" X )  C_  ran  F
2 eqid 2296 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2296 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
42, 3mhmf 14436 . . . . 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 5411 . . . 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 3203 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  C_  ( Base `  N ) )
9 eqid 2296 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 eqid 2296 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
119, 10mhm0 14439 . . . 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 5405 . . . . 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 14443 . . . . 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 14445 . . . . 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 5772 . . . 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 2371 . 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 3194 . . . . . . . . 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 3194 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  ( Base `  M )
)
28 eqid 2296 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
29 eqid 2296 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  N )
302, 28, 29mhmlin 14438 . . . . . . . . 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 14446 . . . . . . . . . . 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 5772 . . . . . . . . 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 2371 . . . . . . 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 2639 . . . . 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 5882 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) ) )
4241eleq1d 2362 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) ) )
4342ralima 5774 . . . . . . 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 2639 . . 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 5881 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) y ) )
4948eleq1d 2362 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) y )  e.  ( F " X ) ) )
5049ralbidv 2576 . . . . 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 5774 . . . 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 14435 . . . 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 14441 . . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   MndHom cmhm 14429  SubMndcsubmnd 14430
This theorem is referenced by:  rhmima  15592
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-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-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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-submnd 14432
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