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Theorem resmhm2b 14438
Description: Restriction of a 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 14418 . . . . 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 14431 . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
8 eqid 2283 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
97, 8mhmf 14420 . . . . . . . 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 5389 . . . . . . 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 5259 . . . . . 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 14432 . . . . . . 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 5377 . . . . . 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 2283 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
22 eqid 2283 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
237, 21, 22mhmlin 14422 . . . . . . . 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 13250 . . . . . . . 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 5875 . . . . . 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 2315 . . . . 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 2635 . . . 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 2283 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
32 eqid 2283 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
3331, 32mhm0 14423 . . . . . 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 14433 . . . . . 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 2315 . . . 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 2283 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
40 eqid 2283 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
41 eqid 2283 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
427, 39, 21, 40, 31, 41ismhm 14417 . . 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 14437 . . . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ran crn 4690    Fn wfn 5250   -->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:  resghm2b  14701  dchrghm  20495  lgseisenlem4  20591
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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