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Theorem resmhm 15969
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 15949 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
2 resmhm.u . . . 4  |-  U  =  ( Ss  X )
32submmnd 15964 . . 3  |-  ( X  e.  (SubMnd `  S
)  ->  U  e.  Mnd )
41, 3anim12ci 567 . 2  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( U  e.  Mnd  /\  T  e. 
Mnd ) )
5 eqid 2443 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
6 eqid 2443 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
75, 6mhmf 15950 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
85submss 15960 . . . . 5  |-  ( X  e.  (SubMnd `  S
)  ->  X  C_  ( Base `  S ) )
9 fssres 5741 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
107, 8, 9syl2an 477 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
118adantl 466 . . . . . 6  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  C_  ( Base `  S ) )
122, 5ressbas2 14670 . . . . . 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 5708 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1510, 14mpbid 210 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
16 simpll 753 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  F  e.  ( S MndHom  T ) )
178ad2antlr 726 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  X  C_  ( Base `  S ) )
18 simprl 756 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  X )
1917, 18sseldd 3490 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  ( Base `  S )
)
20 simprr 757 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  X )
2117, 20sseldd 3490 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
22 eqid 2443 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
23 eqid 2443 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
245, 22, 23mhmlin 15952 . . . . . . 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 1229 . . . . . 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 15963 . . . . . . . . 9  |-  ( ( X  e.  (SubMnd `  S )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( +g  `  S
) y )  e.  X )
27263expb 1198 . . . . . . . 8  |-  ( ( X  e.  (SubMnd `  S )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
2827adantll 713 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
29 fvres 5870 . . . . . . 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 5870 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
32 fvres 5870 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3331, 32oveqan12d 6300 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3433adantl 466 . . . . . 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 2494 . . . . 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 2864 . . . 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 14721 . . . . . . . . . 10  |-  ( X  e.  (SubMnd `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3837adantl 466 . . . . . . . . 9  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3938oveqd 6298 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4039fveq2d 5860 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4140eqeq1d 2445 . . . . . 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 3054 . . . . 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 3054 . . . 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 210 . . 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 2443 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4645subm0cl 15962 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  e.  X
)
4746adantl 466 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  e.  X
)
48 fvres 5870 . . . . 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 15966 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5150adantl 466 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5251fveq2d 5860 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( ( F  |`  X ) `  ( 0g `  U
) ) )
53 eqid 2443 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
5445, 53mhm0 15953 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5554adantr 465 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5649, 52, 553eqtr3d 2492 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g
`  T ) )
5715, 44, 563jca 1177 . 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 2443 . . 3  |-  ( Base `  U )  =  (
Base `  U )
59 eqid 2443 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
60 eqid 2443 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
6158, 6, 59, 23, 60, 53ismhm 15947 . 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 664 1  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X )  e.  ( U MndHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793    C_ wss 3461    |` cres 4991   -->wf 5574   ` cfv 5578  (class class class)co 6281   Basecbs 14614   ↾s cress 14615   +g cplusg 14679   0gc0g 14819   Mndcmnd 15898   MndHom cmhm 15943  SubMndcsubmnd 15944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-submnd 15946
This theorem is referenced by:  resrhm  17437  dchrghm  23509
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