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Theorem resmhm2b 15494
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 15472 . . . . 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 15487 . . . . 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 2443 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
8 eqid 2443 . . . . . . . . 9  |-  ( Base `  T )  =  (
Base `  T )
97, 8mhmf 15474 . . . . . . . 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 5564 . . . . . . 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 5427 . . . . . 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 15488 . . . . . . 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
) )
18 feq3 5549 . . . . . 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 210 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F : ( Base `  S
) --> ( Base `  U
) )
21 eqid 2443 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
22 eqid 2443 . . . . . . . . 9  |-  ( +g  `  T )  =  ( +g  `  T )
237, 21, 22mhmlin 15476 . . . . . . . 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 1188 . . . . . . 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 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 ) ) )
263, 22ressplusg 14285 . . . . . . . 8  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2726ad3antrrr 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
) )
2827oveqd 6113 . . . . . 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 2475 . . . . 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 2813 . . . 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 2443 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
32 eqid 2443 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
3331, 32mhm0 15477 . . . . . 6  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3433adantl 466 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
353, 32subm0 15489 . . . . . 6  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3635ad2antrr 725 . . . . 5  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( 0g `  T )  =  ( 0g `  U
) )
3734, 36eqtrd 2475 . . . 4  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  U
) )
3820, 30, 373jca 1168 . . 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 2443 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
40 eqid 2443 . . . 4  |-  ( +g  `  U )  =  ( +g  `  U )
41 eqid 2443 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
427, 39, 21, 40, 31, 41ismhm 15471 . . 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 664 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  T ) )  ->  F  e.  ( S MndHom  U ) )
443resmhm2 15493 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )
4544ancoms 453 . . 3  |-  ( ( X  e.  (SubMnd `  T )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4645adantlr 714 . 2  |-  ( ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  /\  F  e.  ( S MndHom  U ) )  ->  F  e.  ( S MndHom  T ) )
4743, 46impbida 828 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 965    = wceq 1369    e. wcel 1756   A.wral 2720    C_ wss 3333   ran crn 4846    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   0gc0g 14383   Mndcmnd 15414   MndHom cmhm 15467  SubMndcsubmnd 15468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-mhm 15469  df-submnd 15470
This theorem is referenced by:  resghm2b  15770  dchrghm  22600  lgseisenlem4  22696
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