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Theorem resmhm 14436
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 14419 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
2 resmhm.u . . . 4  |-  U  =  ( Ss  X )
32submmnd 14431 . . 3  |-  ( X  e.  (SubMnd `  S
)  ->  U  e.  Mnd )
41, 3anim12ci 550 . 2  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( U  e.  Mnd  /\  T  e. 
Mnd ) )
5 eqid 2283 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
6 eqid 2283 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
75, 6mhmf 14420 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
85submss 14427 . . . . 5  |-  ( X  e.  (SubMnd `  S
)  ->  X  C_  ( Base `  S ) )
9 fssres 5408 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
107, 8, 9syl2an 463 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
118adantl 452 . . . . . 6  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  C_  ( Base `  S ) )
122, 5ressbas2 13199 . . . . . 6  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  U )
)
1311, 12syl 15 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  X  =  ( Base `  U )
)
1413feq2d 5380 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1510, 14mpbid 201 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
16 simpll 730 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  F  e.  ( S MndHom  T ) )
178ad2antlr 707 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  X  C_  ( Base `  S ) )
18 simprl 732 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  X )
1917, 18sseldd 3181 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  ( Base `  S )
)
20 simprr 733 . . . . . . . 8  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  X )
2117, 20sseldd 3181 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
22 eqid 2283 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
23 eqid 2283 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
245, 22, 23mhmlin 14422 . . . . . . 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 1182 . . . . . 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 14430 . . . . . . . . 9  |-  ( ( X  e.  (SubMnd `  S )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( +g  `  S
) y )  e.  X )
27263expb 1152 . . . . . . . 8  |-  ( ( X  e.  (SubMnd `  S )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
2827adantll 694 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
29 fvres 5542 . . . . . . 7  |-  ( ( x ( +g  `  S
) y )  e.  X  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
3028, 29syl 15 . . . . . 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 5542 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
32 fvres 5542 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3331, 32oveqan12d 5877 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3433adantl 452 . . . . . 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 2325 . . . . 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 2635 . . . 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 13250 . . . . . . . . . 10  |-  ( X  e.  (SubMnd `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3837adantl 452 . . . . . . . . 9  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3938oveqd 5875 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4039fveq2d 5529 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4140eqeq1d 2291 . . . . . 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 2748 . . . . 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 2748 . . . 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 201 . . 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 2283 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4645subm0cl 14429 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  e.  X
)
4746adantl 452 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  e.  X
)
48 fvres 5542 . . . . 5  |-  ( ( 0g `  S )  e.  X  ->  (
( F  |`  X ) `
 ( 0g `  S ) )  =  ( F `  ( 0g `  S ) ) )
4947, 48syl 15 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( F `
 ( 0g `  S ) ) )
502, 45subm0 14433 . . . . . 6  |-  ( X  e.  (SubMnd `  S
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5150adantl 452 . . . . 5  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( 0g `  S )  =  ( 0g `  U ) )
5251fveq2d 5529 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( ( F  |`  X ) `  ( 0g `  U
) ) )
53 eqid 2283 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
5445, 53mhm0 14423 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5554adantr 451 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
5649, 52, 553eqtr3d 2323 . . 3  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  U ) )  =  ( 0g
`  T ) )
5715, 44, 563jca 1132 . 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 2283 . . 3  |-  ( Base `  U )  =  (
Base `  U )
59 eqid 2283 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
60 eqid 2283 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
6158, 6, 59, 23, 60, 53ismhm 14417 . 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 645 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   MndHom cmhm 14413  SubMndcsubmnd 14414
This theorem is referenced by:  resrhm  15574  dchrghm  20495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-submnd 14416
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