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Theorem mhmima 16111
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 5260 . . 3  |-  ( F
" X )  C_  ran  F
2 eqid 2382 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2382 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
42, 3mhmf 16088 . . . . 5  |-  ( F  e.  ( M MndHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
54adantr 463 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
6 frn 5645 . . . 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 3428 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F " X )  C_  ( Base `  N ) )
9 eqid 2382 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
10 eqid 2382 . . . . 5  |-  ( 0g
`  N )  =  ( 0g `  N
)
119, 10mhm0 16091 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
1211adantr 463 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  =  ( 0g `  N ) )
13 ffn 5639 . . . . 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 16098 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  X  C_  ( Base `  M ) )
1615adantl 464 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  X  C_  ( Base `  M ) )
179subm0cl 16100 . . . . 5  |-  ( X  e.  (SubMnd `  M
)  ->  ( 0g `  M )  e.  X
)
1817adantl 464 . . . 4  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  M )  e.  X
)
19 fnfvima 6051 . . . 4  |-  ( ( F  Fn  ( Base `  M )  /\  X  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  X
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2014, 16, 18, 19syl3anc 1226 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( F `  ( 0g `  M
) )  e.  ( F " X ) )
2112, 20eqeltrrd 2471 . 2  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  ( 0g `  N )  e.  ( F " X ) )
22 simpll 751 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  e.  ( M MndHom  N ) )
2316adantr 463 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  X  C_  ( Base `  M ) )
24 simprl 754 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  X )
2523, 24sseldd 3418 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  z  e.  ( Base `  M )
)
26 simprr 755 . . . . . . . . . 10  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  X )
2723, 26sseldd 3418 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  x  e.  ( Base `  M )
)
28 eqid 2382 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
29 eqid 2382 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  N )
302, 28, 29mhmlin 16090 . . . . . . . . 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 1226 . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  F  Fn  ( Base `  M )
)
3328submcl 16101 . . . . . . . . . . 11  |-  ( ( X  e.  (SubMnd `  M )  /\  z  e.  X  /\  x  e.  X )  ->  (
z ( +g  `  M
) x )  e.  X )
34333expb 1195 . . . . . . . . . 10  |-  ( ( X  e.  (SubMnd `  M )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
3534adantll 711 . . . . . . . . 9  |-  ( ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M
) )  /\  (
z  e.  X  /\  x  e.  X )
)  ->  ( z
( +g  `  M ) x )  e.  X
)
36 fnfvima 6051 . . . . . . . . 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 1226 . . . . . . . 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 2471 . . . . . . 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 646 . . . . . 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 2796 . . . . 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 6204 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( F `  z
) ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) ( F `
 x ) ) )
4241eleq1d 2451 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( ( F `  z ) ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) ( F `  x
) )  e.  ( F " X ) ) )
4342ralima 6053 . . . . . . 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 659 . . . . . 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 463 . . . . 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 2796 . . 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 6203 . . . . . . 7  |-  ( x  =  ( F `  z )  ->  (
x ( +g  `  N
) y )  =  ( ( F `  z ) ( +g  `  N ) y ) )
4948eleq1d 2451 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
( x ( +g  `  N ) y )  e.  ( F " X )  <->  ( ( F `  z )
( +g  `  N ) y )  e.  ( F " X ) ) )
5049ralbidv 2821 . . . . 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 6053 . . . 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 659 . . 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 16087 . . . 4  |-  ( F  e.  ( M MndHom  N
)  ->  N  e.  Mnd )
5554adantr 463 . . 3  |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M )
)  ->  N  e.  Mnd )
563, 10, 29issubm 16095 . . 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 1177 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389   ran crn 4914   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847   Mndcmnd 16036   MndHom cmhm 16081  SubMndcsubmnd 16082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084
This theorem is referenced by:  rhmima  17573
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