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Theorem resmhm2b 15861
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 15838 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  ->  S  e.  Mnd )
21adantl 466 . . . 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 15854 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  U  e.  Mnd )
54ad2antrr 725 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  U  e.  Mnd )
62, 5jca 532 . . 3  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( S  e.  Mnd  /\  U  e.  Mnd ) )
7 eqid 2441 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
8 eqid 2441 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
97, 8mhmf 15840 . . . . . . . 8  |-  ( F  e.  ( S MndHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
109adantl 466 . . . . . . 7  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  T
) )
11 ffn 5717 . . . . . . 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 754 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ran  F 
C_  X )
14 df-f 5578 . . . . . 6  |-  ( F : ( Base `  S
) --> X  <->  ( F  Fn  ( Base `  S
)  /\  ran  F  C_  X ) )
1512, 13, 14sylanbrc 664 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> X )
163submbas 15855 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
1716ad2antrr 725 . . . . . 6  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  X  =  ( Base `  U
) )
1817feq3d 5705 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F : ( Base `  S
) --> X  <->  F :
( Base `  S ) --> ( Base `  U )
) )
1915, 18mpbid 210 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  U
) )
20 eqid 2441 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
21 eqid 2441 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
227, 20, 21mhmlin 15842 . . . . . . . 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 ) ) )
23223expb 1196 . . . . . . 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
) ) )
2423adantll 713 . . . . . 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 ) ) )
253, 21ressplusg 14611 . . . . . . . 8  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2625ad3antrrr 729 . . . . . . 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
) )
2726oveqd 6294 . . . . . 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 ) ) )
2824, 27eqtrd 2482 . . . . 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 ) ) )
2928ralrimivva 2862 . . . 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 ) ) )
30 eqid 2441 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
31 eqid 2441 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
3230, 31mhm0 15843 . . . . . 6  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3332adantl 466 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
343, 31subm0 15856 . . . . . 6  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3534ad2antrr 725 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( 0g `  T )  =  ( 0g `  U
) )
3633, 35eqtrd 2482 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  U
) )
3719, 29, 363jca 1175 . . 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 ) ) )
38 eqid 2441 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
39 eqid 2441 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
40 eqid 2441 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
417, 38, 20, 39, 30, 40ismhm 15837 . . 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 ) ) ) )
426, 37, 41sylanbrc 664 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  e.  ( S MndHom  U ) )
433resmhm2 15860 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )
4443ancoms 453 . . 3  |-  ( ( X  e.  (SubMnd `  T )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4544adantlr 714 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4642, 45impbida 830 1  |-  ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   ran crn 4986    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   0gc0g 14709   Mndcmnd 15788   MndHom cmhm 15833  SubMndcsubmnd 15834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-submnd 15836
This theorem is referenced by:  resghm2b  16154  m2cpmmhm  19113  dchrghm  23396  lgseisenlem4  23492
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