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Theorem resmhm2b 14454
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 14434 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
21adantl 452 . . . 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 14447 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  U  e.  Mnd )
54ad2antrr 706 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  U  e.  Mnd )
62, 5jca 518 . . 3  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( S  e.  Mnd  /\  U  e.  Mnd ) )
7 eqid 2296 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
8 eqid 2296 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
97, 8mhmf 14436 . . . . . . . 8  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
109adantl 452 . . . . . . 7  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  T
) )
11 ffn 5405 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1210, 11syl 15 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  Fn  ( Base `  S
) )
13 simplr 731 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ran  F 
C_  X )
14 df-f 5275 . . . . . 6  |-  ( F : ( Base `  S
) --> X  <->  ( F  Fn  ( Base `  S
)  /\  ran  F  C_  X ) )
1512, 13, 14sylanbrc 645 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> X )
163submbas 14448 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
1716ad2antrr 706 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  X  =  ( Base `  U
) )
18 feq3 5393 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( F : ( Base `  S
) --> X  <->  F :
( Base `  S ) --> ( Base `  U )
) )
1917, 18syl 15 . . . . 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 201 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  U
) )
21 eqid 2296 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
22 eqid 2296 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
237, 21, 22mhmlin 14438 . . . . . . . 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 1152 . . . . . . 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 694 . . . . . 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 13266 . . . . . . . 8  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2726ad3antrrr 710 . . . . . . 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 5891 . . . . . 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 2328 . . . . 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 2648 . . . 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 2296 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
32 eqid 2296 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
3331, 32mhm0 14439 . . . . . 6  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3433adantl 452 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
353, 32subm0 14449 . . . . . 6  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3635ad2antrr 706 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( 0g `  T )  =  ( 0g `  U
) )
3734, 36eqtrd 2328 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  U
) )
3820, 30, 373jca 1132 . . 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 2296 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
40 eqid 2296 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
41 eqid 2296 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
427, 39, 21, 40, 31, 41ismhm 14433 . . 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 645 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  e.  ( S MndHom  U ) )
443resmhm2 14453 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )
4544ancoms 439 . . 3  |-  ( ( X  e.  (SubMnd `  T )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4645adantlr 695 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4743, 46impbida 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   MndHom cmhm 14429  SubMndcsubmnd 14430
This theorem is referenced by:  resghm2b  14717  dchrghm  20511  lgseisenlem4  20607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-submnd 14432
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