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Theorem mhmima 15804
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 5346 . . 3  |-  ( F
" X )  C_  ran  F
2 eqid 2467 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2467 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
42, 3mhmf 15782 . . . . 5  |-  ( F  e.  ( M MndHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
54adantr 465 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
6 frn 5735 . . . 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 3515 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  C_  ( Base `  N ) )
9 eqid 2467 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 eqid 2467 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
119, 10mhm0 15785 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
1211adantr 465 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
13 ffn 5729 . . . . 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 15791 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  X  C_  ( Base `  M ) )
1615adantl 466 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  X  C_  ( Base `  M ) )
179subm0cl 15793 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  ( 0g `  M )  e.  X
)
1817adantl 466 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  M )  e.  X
)
19 fnfvima 6136 . . . 4  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  X
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2014, 16, 18, 19syl3anc 1228 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2112, 20eqeltrrd 2556 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  N )  e.  ( F " X ) )
22 simpll 753 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  e.  ( M MndHom  N ) )
2316adantr 465 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  X  C_  ( Base `  M ) )
24 simprl 755 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  X )
2523, 24sseldd 3505 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  ( Base `  M )
)
26 simprr 756 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  X )
2723, 26sseldd 3505 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  ( Base `  M )
)
28 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
29 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  N )
302, 28, 29mhmlin 15784 . . . . . . . . 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 1228 . . . . . . . 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 465 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  Fn  ( Base `  M )
)
3328submcl 15794 . . . . . . . . . . 11  |-  ( ( X  e.  (SubMnd `  M )  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( +g  `  M
) x )  e.  X )
34333expb 1197 . . . . . . . . . 10  |-  ( ( X  e.  (SubMnd `  M )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
3534adantll 713 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
36 fnfvima 6136 . . . . . . . . 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 1228 . . . . . . . 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 2556 . . . . . . 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 648 . . . . . 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 2878 . . . . 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 6290 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) ) )
4241eleq1d 2536 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) ) )
4342ralima 6138 . . . . . . 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 661 . . . . . 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 465 . . . . 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 232 . . . 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 2878 . . 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 6289 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) y ) )
4948eleq1d 2536 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) y )  e.  ( F " X ) ) )
5049ralbidv 2903 . . . . 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 6138 . . . 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 661 . . 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 232 . 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 15781 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5554adantr 465 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  N  e.  Mnd )
563, 10, 29issubm 15788 . . 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 1179 1  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  e.  (SubMnd `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Mndcmnd 15722   MndHom cmhm 15775  SubMndcsubmnd 15776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-mnd 15728  df-mhm 15777  df-submnd 15778
This theorem is referenced by:  rhmima  17243
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