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Theorem resmhm 14687
Description: Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypothesis
Ref Expression
resmhm.u  |-  U  =  ( Ss  X )
Assertion
Ref Expression
resmhm  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X )  e.  ( U MndHom  T ) )

Proof of Theorem resmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl2 14670 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
2 resmhm.u . . . 4  |-  U  =  ( Ss  X )
32submmnd 14682 . . 3  |-  ( X  e.  (SubMnd `  S
)  ->  U  e.  Mnd )
41, 3anim12ci 551 . 2  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( U  e.  Mnd  /\  T  e. 
Mnd ) )
5 eqid 2388 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
6 eqid 2388 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
75, 6mhmf 14671 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
85submss 14678 . . . . 5  |-  ( X  e.  (SubMnd `  S
)  ->  X  C_  ( Base `  S ) )
9 fssres 5551 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
107, 8, 9syl2an 464 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
118adantl 453 . . . . . 6  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  C_  ( Base `  S ) )
122, 5ressbas2 13448 . . . . . 6  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  U )
)
1311, 12syl 16 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  =  ( Base `  U )
)
1413feq2d 5522 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1510, 14mpbid 202 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
16 simpll 731 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  F  e.  ( S MndHom  T ) )
178ad2antlr 708 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  X  C_  ( Base `  S ) )
18 simprl 733 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  X )
1917, 18sseldd 3293 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  ( Base `  S )
)
20 simprr 734 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  X )
2117, 20sseldd 3293 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
22 eqid 2388 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
23 eqid 2388 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
245, 22, 23mhmlin 14673 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
2516, 19, 21, 24syl3anc 1184 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
2622submcl 14681 . . . . . . . . 9  |-  ( ( X  e.  (SubMnd `  S )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( +g  `  S
) y )  e.  X )
27263expb 1154 . . . . . . . 8  |-  ( ( X  e.  (SubMnd `  S )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
2827adantll 695 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
29 fvres 5686 . . . . . . 7  |-  ( ( x ( +g  `  S
) y )  e.  X  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
3028, 29syl 16 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
31 fvres 5686 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
32 fvres 5686 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3331, 32oveqan12d 6040 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3433adantl 453 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3525, 30, 343eqtr4d 2430 . . . . 5  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) ) )
3635ralrimivva 2742 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  A. x  e.  X  A. y  e.  X  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) ) )
372, 22ressplusg 13499 . . . . . . . . . 10  |-  ( X  e.  (SubMnd `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3837adantl 453 . . . . . . . . 9  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3938oveqd 6038 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4039fveq2d 5673 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4140eqeq1d 2396 . . . . . 6  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( (
( F  |`  X ) `
 ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  <->  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) )  =  ( ( ( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) ) ) )
4213, 41raleqbidv 2860 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( A. y  e.  X  (
( F  |`  X ) `
 ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  <->  A. y  e.  ( Base `  U
) ( ( F  |`  X ) `  (
x ( +g  `  U
) y ) )  =  ( ( ( F  |`  X ) `  x ) ( +g  `  T ) ( ( F  |`  X ) `  y ) ) ) )
4313, 42raleqbidv 2860 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( A. x  e.  X  A. y  e.  X  (
( F  |`  X ) `
 ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  <->  A. x  e.  ( Base `  U
) A. y  e.  ( Base `  U
) ( ( F  |`  X ) `  (
x ( +g  `  U
) y ) )  =  ( ( ( F  |`  X ) `  x ) ( +g  `  T ) ( ( F  |`  X ) `  y ) ) ) )
4436, 43mpbid 202 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  A. x  e.  ( Base `  U
) A. y  e.  ( Base `  U
) ( ( F  |`  X ) `  (
x ( +g  `  U
) y ) )  =  ( ( ( F  |`  X ) `  x ) ( +g  `  T ) ( ( F  |`  X ) `  y ) ) )
45 eqid 2388 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4645subm0cl 14680 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  e.  X
)
4746adantl 453 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  e.  X
)
48 fvres 5686 . . . . 5  |-  ( ( 0g `  S )  e.  X  ->  (
( F  |`  X ) `
 ( 0g `  S ) )  =  ( F `  ( 0g `  S ) ) )
4947, 48syl 16 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( F `
 ( 0g `  S ) ) )
502, 45subm0 14684 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5150adantl 453 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5251fveq2d 5673 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( ( F  |`  X ) `  ( 0g `  U
) ) )
53 eqid 2388 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
5445, 53mhm0 14674 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5554adantr 452 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5649, 52, 553eqtr3d 2428 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g
`  T ) )
5715, 44, 563jca 1134 . 2  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) : ( Base `  U
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  U
) A. y  e.  ( Base `  U
) ( ( F  |`  X ) `  (
x ( +g  `  U
) y ) )  =  ( ( ( F  |`  X ) `  x ) ( +g  `  T ) ( ( F  |`  X ) `  y ) )  /\  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g `  T ) ) )
58 eqid 2388 . . 3  |-  ( Base `  U )  =  (
Base `  U )
59 eqid 2388 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
60 eqid 2388 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
6158, 6, 59, 23, 60, 53ismhm 14668 . 2  |-  ( ( F  |`  X )  e.  ( U MndHom  T )  <-> 
( ( U  e. 
Mnd  /\  T  e.  Mnd )  /\  (
( F  |`  X ) : ( Base `  U
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  U
) A. y  e.  ( Base `  U
) ( ( F  |`  X ) `  (
x ( +g  `  U
) y ) )  =  ( ( ( F  |`  X ) `  x ) ( +g  `  T ) ( ( F  |`  X ) `  y ) )  /\  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g `  T ) ) ) )
624, 57, 61sylanbrc 646 1  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X )  e.  ( U MndHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650    C_ wss 3264    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397   ↾s cress 13398   +g cplusg 13457   0gc0g 13651   Mndcmnd 14612   MndHom cmhm 14664  SubMndcsubmnd 14665
This theorem is referenced by:  resrhm  15825  dchrghm  20908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-mhm 14666  df-submnd 14667
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