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Theorem mhmima 14763
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 5216 . . 3  |-  ( F
" X )  C_  ran  F
2 eqid 2436 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2436 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
42, 3mhmf 14743 . . . . 5  |-  ( F  e.  ( M MndHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
54adantr 452 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
6 frn 5597 . . . 4  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  ran  F  C_  ( Base `  N )
)
75, 6syl 16 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ran  F  C_  ( Base `  N )
)
81, 7syl5ss 3359 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  C_  ( Base `  N ) )
9 eqid 2436 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 eqid 2436 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
119, 10mhm0 14746 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
1211adantr 452 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
13 ffn 5591 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
145, 13syl 16 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F  Fn  ( Base `  M )
)
152submss 14750 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  X  C_  ( Base `  M ) )
1615adantl 453 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  X  C_  ( Base `  M ) )
179subm0cl 14752 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  ( 0g `  M )  e.  X
)
1817adantl 453 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  M )  e.  X
)
19 fnfvima 5976 . . . 4  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  X
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2014, 16, 18, 19syl3anc 1184 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2112, 20eqeltrrd 2511 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  N )  e.  ( F " X ) )
22 simpll 731 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  e.  ( M MndHom  N ) )
2316adantr 452 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  X  C_  ( Base `  M ) )
24 simprl 733 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  X )
2523, 24sseldd 3349 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  ( Base `  M )
)
26 simprr 734 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  X )
2723, 26sseldd 3349 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  ( Base `  M )
)
28 eqid 2436 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
29 eqid 2436 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  N )
302, 28, 29mhmlin 14745 . . . . . . . . 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 1184 . . . . . . . 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 452 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  Fn  ( Base `  M )
)
3328submcl 14753 . . . . . . . . . . 11  |-  ( ( X  e.  (SubMnd `  M )  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( +g  `  M
) x )  e.  X )
34333expb 1154 . . . . . . . . . 10  |-  ( ( X  e.  (SubMnd `  M )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
3534adantll 695 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
36 fnfvima 5976 . . . . . . . . 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 1184 . . . . . . . 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 2511 . . . . . . 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 630 . . . . . 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 2789 . . . . 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 6089 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) ) )
4241eleq1d 2502 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) ) )
4342ralima 5978 . . . . . . 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 643 . . . . . 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 452 . . . . 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 224 . . . 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 2789 . . 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 6088 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) y ) )
4948eleq1d 2502 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) y )  e.  ( F " X ) ) )
5049ralbidv 2725 . . . . 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 5978 . . . 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 643 . . 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 224 . 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 14742 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5554adantr 452 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  N  e.  Mnd )
563, 10, 29issubm 14748 . . 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 16 . 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 1137 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684   MndHom cmhm 14736  SubMndcsubmnd 14737
This theorem is referenced by:  rhmima  15899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-mhm 14738  df-submnd 14739
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