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Theorem resmhm 16606
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 16586 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
2 resmhm.u . . . 4  |-  U  =  ( Ss  X )
32submmnd 16601 . . 3  |-  ( X  e.  (SubMnd `  S
)  ->  U  e.  Mnd )
41, 3anim12ci 571 . 2  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( U  e.  Mnd  /\  T  e. 
Mnd ) )
5 eqid 2451 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
6 eqid 2451 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
75, 6mhmf 16587 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
85submss 16597 . . . . 5  |-  ( X  e.  (SubMnd `  S
)  ->  X  C_  ( Base `  S ) )
9 fssres 5749 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
107, 8, 9syl2an 480 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
118adantl 468 . . . . . 6  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  C_  ( Base `  S ) )
122, 5ressbas2 15180 . . . . . 6  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  U )
)
1311, 12syl 17 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  =  ( Base `  U )
)
1413feq2d 5715 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1510, 14mpbid 214 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
16 simpll 760 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  F  e.  ( S MndHom  T ) )
178ad2antlr 733 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  X  C_  ( Base `  S ) )
18 simprl 764 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  X )
1917, 18sseldd 3433 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  ( Base `  S )
)
20 simprr 766 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  X )
2117, 20sseldd 3433 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
22 eqid 2451 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
23 eqid 2451 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
245, 22, 23mhmlin 16589 . . . . . . 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 1268 . . . . . 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 16600 . . . . . . . . 9  |-  ( ( X  e.  (SubMnd `  S )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( +g  `  S
) y )  e.  X )
27263expb 1209 . . . . . . . 8  |-  ( ( X  e.  (SubMnd `  S )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
2827adantll 720 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
29 fvres 5879 . . . . . . 7  |-  ( ( x ( +g  `  S
) y )  e.  X  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
3028, 29syl 17 . . . . . 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 5879 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
32 fvres 5879 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3331, 32oveqan12d 6309 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3433adantl 468 . . . . . 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 2495 . . . . 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 2809 . . . 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 15239 . . . . . . . . . 10  |-  ( X  e.  (SubMnd `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3837adantl 468 . . . . . . . . 9  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3938oveqd 6307 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4039fveq2d 5869 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4140eqeq1d 2453 . . . . . 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 3001 . . . . 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 3001 . . . 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 214 . . 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 2451 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4645subm0cl 16599 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  e.  X
)
4746adantl 468 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  e.  X
)
48 fvres 5879 . . . . 5  |-  ( ( 0g `  S )  e.  X  ->  (
( F  |`  X ) `
 ( 0g `  S ) )  =  ( F `  ( 0g `  S ) ) )
4947, 48syl 17 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( F `
 ( 0g `  S ) ) )
502, 45subm0 16603 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5150adantl 468 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5251fveq2d 5869 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( ( F  |`  X ) `  ( 0g `  U
) ) )
53 eqid 2451 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
5445, 53mhm0 16590 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5554adantr 467 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5649, 52, 553eqtr3d 2493 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g
`  T ) )
5715, 44, 563jca 1188 . 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 2451 . . 3  |-  ( Base `  U )  =  (
Base `  U )
59 eqid 2451 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
60 eqid 2451 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
6158, 6, 59, 23, 60, 53ismhm 16584 . 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 670 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   Basecbs 15121   ↾s cress 15122   +g cplusg 15190   0gc0g 15338   Mndcmnd 16535   MndHom cmhm 16580  SubMndcsubmnd 16581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583
This theorem is referenced by:  resrhm  18037  dchrghm  24184
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