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Theorem resmhm2 15587
Description: One direction of resmhm2b 15588. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u  |-  U  =  ( Ts  X )
Assertion
Ref Expression
resmhm2  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )

Proof of Theorem resmhm2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 15566 . . 3  |-  ( F  e.  ( S MndHom  U
)  ->  S  e.  Mnd )
2 submrcl 15573 . . 3  |-  ( X  e.  (SubMnd `  T
)  ->  T  e.  Mnd )
31, 2anim12i 566 . 2  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
4 eqid 2451 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2451 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
64, 5mhmf 15568 . . . 4  |-  ( F  e.  ( S MndHom  U
)  ->  F :
( Base `  S ) --> ( Base `  U )
)
7 resmhm2.u . . . . . 6  |-  U  =  ( Ts  X )
87submbas 15582 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
9 eqid 2451 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
109submss 15577 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  C_  ( Base `  T ) )
118, 10eqsstr3d 3486 . . . 4  |-  ( X  e.  (SubMnd `  T
)  ->  ( Base `  U )  C_  ( Base `  T ) )
12 fss 5662 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  U )  /\  ( Base `  U )  C_  ( Base `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
136, 11, 12syl2an 477 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
14 eqid 2451 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
15 eqid 2451 . . . . . . . 8  |-  ( +g  `  U )  =  ( +g  `  U )
164, 14, 15mhmlin 15570 . . . . . . 7  |-  ( ( F  e.  ( S MndHom  U )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  U
) ( F `  y ) ) )
17163expb 1189 . . . . . 6  |-  ( ( F  e.  ( S MndHom  U )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( F `  (
x ( +g  `  S
) y ) )  =  ( ( F `
 x ) ( +g  `  U ) ( F `  y
) ) )
1817adantlr 714 . . . . 5  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( F `  (
x ( +g  `  S
) y ) )  =  ( ( F `
 x ) ( +g  `  U ) ( F `  y
) ) )
19 eqid 2451 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
207, 19ressplusg 14379 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2120ad2antlr 726 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( +g  `  T )  =  ( +g  `  U
) )
2221oveqd 6204 . . . . 5  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( ( F `  x ) ( +g  `  T ) ( F `
 y ) )  =  ( ( F `
 x ) ( +g  `  U ) ( F `  y
) ) )
2318, 22eqtr4d 2494 . . . 4  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( F `  (
x ( +g  `  S
) y ) )  =  ( ( F `
 x ) ( +g  `  T ) ( F `  y
) ) )
2423ralrimivva 2901 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
25 eqid 2451 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
26 eqid 2451 . . . . . 6  |-  ( 0g
`  U )  =  ( 0g `  U
)
2725, 26mhm0 15571 . . . . 5  |-  ( F  e.  ( S MndHom  U
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
2827adantr 465 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
29 eqid 2451 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
307, 29subm0 15583 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3130adantl 466 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3228, 31eqtr4d 2494 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3313, 24, 323jca 1168 . 2  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) )  /\  ( F `  ( 0g
`  S ) )  =  ( 0g `  T ) ) )
344, 9, 14, 19, 25, 29ismhm 15565 . 2  |-  ( F  e.  ( S MndHom  T
)  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) )  /\  ( F `  ( 0g
`  S ) )  =  ( 0g `  T ) ) ) )
353, 33, 34sylanbrc 664 1  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F  e.  ( S MndHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2793    C_ wss 3423   -->wf 5509   ` cfv 5513  (class class class)co 6187   Basecbs 14273   ↾s cress 14274   +g cplusg 14337   0gc0g 14477   Mndcmnd 15508   MndHom cmhm 15561  SubMndcsubmnd 15562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-recs 6929  df-rdg 6963  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-plusg 14350  df-0g 14479  df-mnd 15514  df-mhm 15563  df-submnd 15564
This theorem is referenced by:  resmhm2b  15588  resghm2  15863  zrhpsgnmhm  18120  lgseisenlem4  22804
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