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Theorem resmhm2b 14763
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u  |-  U  =  ( Ts  X )
Assertion
Ref Expression
resmhm2b  |-  ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )

Proof of Theorem resmhm2b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 14743 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
21adantl 454 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  S  e.  Mnd )
3 resmhm2.u . . . . . 6  |-  U  =  ( Ts  X )
43submmnd 14756 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  U  e.  Mnd )
54ad2antrr 708 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  U  e.  Mnd )
62, 5jca 520 . . 3  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( S  e.  Mnd  /\  U  e.  Mnd ) )
7 eqid 2438 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
8 eqid 2438 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
97, 8mhmf 14745 . . . . . . . 8  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
109adantl 454 . . . . . . 7  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  T
) )
11 ffn 5593 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1210, 11syl 16 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  Fn  ( Base `  S
) )
13 simplr 733 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ran  F 
C_  X )
14 df-f 5460 . . . . . 6  |-  ( F : ( Base `  S
) --> X  <->  ( F  Fn  ( Base `  S
)  /\  ran  F  C_  X ) )
1512, 13, 14sylanbrc 647 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> X )
163submbas 14757 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
1716ad2antrr 708 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  X  =  ( Base `  U
) )
18 feq3 5580 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( F : ( Base `  S
) --> X  <->  F :
( Base `  S ) --> ( Base `  U )
) )
1917, 18syl 16 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F : ( Base `  S
) --> X  <->  F :
( Base `  S ) --> ( Base `  U )
) )
2015, 19mpbid 203 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  U
) )
21 eqid 2438 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
22 eqid 2438 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
237, 21, 22mhmlin 14747 . . . . . . . 8  |-  ( ( F  e.  ( S MndHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
24233expb 1155 . . . . . . 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
) ) )
2524adantll 696 . . . . . 6  |-  ( ( ( ( X  e.  (SubMnd `  T )  /\  ran  F  C_  X
)  /\  F  e.  ( S MndHom  T ) )  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) )
263, 22ressplusg 13573 . . . . . . . 8  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2726ad3antrrr 712 . . . . . . 7  |-  ( ( ( ( X  e.  (SubMnd `  T )  /\  ran  F  C_  X
)  /\  F  e.  ( S MndHom  T ) )  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  ( +g  `  T )  =  ( +g  `  U
) )
2827oveqd 6100 . . . . . 6  |-  ( ( ( ( X  e.  (SubMnd `  T )  /\  ran  F  C_  X
)  /\  F  e.  ( S MndHom  T ) )  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  (
( F `  x
) ( +g  `  T
) ( F `  y ) )  =  ( ( F `  x ) ( +g  `  U ) ( F `
 y ) ) )
2925, 28eqtrd 2470 . . . . 5  |-  ( ( ( ( X  e.  (SubMnd `  T )  /\  ran  F  C_  X
)  /\  F  e.  ( S MndHom  T ) )  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  U ) ( F `
 y ) ) )
3029ralrimivva 2800 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  U
) ( F `  y ) ) )
31 eqid 2438 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
32 eqid 2438 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
3331, 32mhm0 14748 . . . . . 6  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3433adantl 454 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
353, 32subm0 14758 . . . . . 6  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3635ad2antrr 708 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( 0g `  T )  =  ( 0g `  U
) )
3734, 36eqtrd 2470 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  U
) )
3820, 30, 373jca 1135 . . 3  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F : ( Base `  S
) --> ( Base `  U
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  U
) ( F `  y ) )  /\  ( F `  ( 0g
`  S ) )  =  ( 0g `  U ) ) )
39 eqid 2438 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
40 eqid 2438 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
41 eqid 2438 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
427, 39, 21, 40, 31, 41ismhm 14742 . . 3  |-  ( F  e.  ( S MndHom  U
)  <->  ( ( S  e.  Mnd  /\  U  e.  Mnd )  /\  ( F : ( Base `  S
) --> ( Base `  U
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  U
) ( F `  y ) )  /\  ( F `  ( 0g
`  S ) )  =  ( 0g `  U ) ) ) )
436, 38, 42sylanbrc 647 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  e.  ( S MndHom  U ) )
443resmhm2 14762 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )
4544ancoms 441 . . 3  |-  ( ( X  e.  (SubMnd `  T )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4645adantlr 697 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4743, 46impbida 807 1  |-  ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ran crn 4881    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686   MndHom cmhm 14738  SubMndcsubmnd 14739
This theorem is referenced by:  resghm2b  15026  dchrghm  21042  lgseisenlem4  21138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-mhm 14740  df-submnd 14741
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