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Theorem resmgmhm 39785
Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
resmgmhm.u  |-  U  =  ( Ss  X )
Assertion
Ref Expression
resmgmhm  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( F  |`  X )  e.  ( U MgmHom  T ) )

Proof of Theorem resmgmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 39768 . . . 4  |-  ( F  e.  ( S MgmHom  T
)  ->  ( S  e. Mgm  /\  T  e. Mgm )
)
21simprd 465 . . 3  |-  ( F  e.  ( S MgmHom  T
)  ->  T  e. Mgm )
3 resmgmhm.u . . . 4  |-  U  =  ( Ss  X )
43submgmmgm 39782 . . 3  |-  ( X  e.  (SubMgm `  S
)  ->  U  e. Mgm )
52, 4anim12ci 570 . 2  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( U  e. Mgm  /\  T  e. Mgm )
)
6 eqid 2450 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
7 eqid 2450 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
86, 7mgmhmf 39771 . . . . 5  |-  ( F  e.  ( S MgmHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
96submgmss 39779 . . . . 5  |-  ( X  e.  (SubMgm `  S
)  ->  X  C_  ( Base `  S ) )
10 fssres 5747 . . . . 5  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
118, 9, 10syl2an 480 . . . 4  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
129adantl 468 . . . . . 6  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  X  C_  ( Base `  S ) )
133, 6ressbas2 15173 . . . . . 6  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  U )
)
1412, 13syl 17 . . . . 5  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  X  =  ( Base `  U )
)
1514feq2d 5713 . . . 4  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1611, 15mpbid 214 . . 3  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
17 simpll 759 . . . . . . 7  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  F  e.  ( S MgmHom  T ) )
189ad2antlr 732 . . . . . . . 8  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  X  C_  ( Base `  S ) )
19 simprl 763 . . . . . . . 8  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  X )
2018, 19sseldd 3432 . . . . . . 7  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  x  e.  ( Base `  S )
)
21 simprr 765 . . . . . . . 8  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  X )
2218, 21sseldd 3432 . . . . . . 7  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  y  e.  ( Base `  S )
)
23 eqid 2450 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
24 eqid 2450 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
256, 23, 24mgmhmlin 39773 . . . . . . 7  |-  ( ( F  e.  ( S MgmHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
2617, 20, 22, 25syl3anc 1267 . . . . . 6  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
2723submgmcl 39781 . . . . . . . . 9  |-  ( ( X  e.  (SubMgm `  S )  /\  x  e.  X  /\  y  e.  X )  ->  (
x ( +g  `  S
) y )  e.  X )
28273expb 1208 . . . . . . . 8  |-  ( ( X  e.  (SubMgm `  S )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
2928adantll 719 . . . . . . 7  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( x
( +g  `  S ) y )  e.  X
)
30 fvres 5877 . . . . . . 7  |-  ( ( x ( +g  `  S
) y )  e.  X  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
3129, 30syl 17 . . . . . 6  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( F `
 ( x ( +g  `  S ) y ) ) )
32 fvres 5877 . . . . . . . 8  |-  ( x  e.  X  ->  (
( F  |`  X ) `
 x )  =  ( F `  x
) )
33 fvres 5877 . . . . . . . 8  |-  ( y  e.  X  ->  (
( F  |`  X ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 6307 . . . . . . 7  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( ( ( F  |`  X ) `  x
) ( +g  `  T
) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3534adantl 468 . . . . . 6  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( (
( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) )  =  ( ( F `  x
) ( +g  `  T
) ( F `  y ) ) )
3626, 31, 353eqtr4d 2494 . . . . 5  |-  ( ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S
) )  /\  (
x  e.  X  /\  y  e.  X )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) ) )
3736ralrimivva 2808 . . . 4  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  A. x  e.  X  A. y  e.  X  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( ( F  |`  X ) `
 x ) ( +g  `  T ) ( ( F  |`  X ) `  y
) ) )
383, 23ressplusg 15232 . . . . . . . . . 10  |-  ( X  e.  (SubMgm `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3938adantl 468 . . . . . . . . 9  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
4039oveqd 6305 . . . . . . . 8  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( x
( +g  `  S ) y )  =  ( x ( +g  `  U
) y ) )
4140fveq2d 5867 . . . . . . 7  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( ( F  |`  X ) `  ( x ( +g  `  S ) y ) )  =  ( ( F  |`  X ) `  ( x ( +g  `  U ) y ) ) )
4241eqeq1d 2452 . . . . . 6  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  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
) ) ) )
4314, 42raleqbidv 3000 . . . . 5  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  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 ) ) ) )
4414, 43raleqbidv 3000 . . . 4  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  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 ) ) ) )
4537, 44mpbid 214 . . 3  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  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 ) ) )
4616, 45jca 535 . 2  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  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 ) ) ) )
47 eqid 2450 . . 3  |-  ( Base `  U )  =  (
Base `  U )
48 eqid 2450 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
4947, 7, 48, 24ismgmhm 39770 . 2  |-  ( ( F  |`  X )  e.  ( U MgmHom  T )  <-> 
( ( U  e. Mgm  /\  T  e. Mgm )  /\  ( ( 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
) ) ) ) )
505, 46, 49sylanbrc 669 1  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S )
)  ->  ( F  |`  X )  e.  ( U MgmHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736    C_ wss 3403    |` cres 4835   -->wf 5577   ` cfv 5581  (class class class)co 6288   Basecbs 15114   ↾s cress 15115   +g cplusg 15183  Mgmcmgm 16479   MgmHom cmgmhm 39764  SubMgmcsubmgm 39765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mgm 16481  df-mgmhm 39766  df-submgm 39767
This theorem is referenced by: (None)
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