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Theorem resmhm 14751
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 14734 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  T  e.  Mnd )
2 resmhm.u . . . 4  |-  U  =  ( Ss  X )
32submmnd 14746 . . 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 2435 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
6 eqid 2435 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
75, 6mhmf 14735 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
85submss 14742 . . . . 5  |-  ( X  e.  (SubMnd `  S
)  ->  X  C_  ( Base `  S ) )
9 fssres 5602 . . . . 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 13512 . . . . . 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 5573 . . . 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 3341 . . . . . . 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 3341 . . . . . . 7  |-  ( ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
22 eqid 2435 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
23 eqid 2435 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
245, 22, 23mhmlin 14737 . . . . . . 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 14745 . . . . . . . . 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 5737 . . . . . . 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 5737 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
32 fvres 5737 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3331, 32oveqan12d 6092 . . . . . . 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 2477 . . . . 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 2790 . . . 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 13563 . . . . . . . . . 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 6090 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4039fveq2d 5724 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4140eqeq1d 2443 . . . . . 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 2908 . . . . 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 2908 . . . 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 2435 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4645subm0cl 14744 . . . . . 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 5737 . . . . 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 14748 . . . . . 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 5724 . . . 4  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S )
)  ->  ( ( F  |`  X ) `  ( 0g `  S ) )  =  ( ( F  |`  X ) `  ( 0g `  U
) ) )
53 eqid 2435 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
5445, 53mhm0 14738 . . . . 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 2475 . . 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 2435 . . 3  |-  ( Base `  U )  =  (
Base `  U )
59 eqid 2435 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
60 eqid 2435 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
6158, 6, 59, 23, 60, 53ismhm 14732 . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   0gc0g 13715   Mndcmnd 14676   MndHom cmhm 14728  SubMndcsubmnd 14729
This theorem is referenced by:  resrhm  15889  dchrghm  21032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-mhm 14730  df-submnd 14731
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