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Theorem resmhm2 16685
Description: One direction of resmhm2b 16686. (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 16663 . . 3  |-  ( F  e.  ( S MndHom  U
)  ->  S  e.  Mnd )
2 submrcl 16671 . . 3  |-  ( X  e.  (SubMnd `  T
)  ->  T  e.  Mnd )
31, 2anim12i 576 . 2  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
4 eqid 2471 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2471 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
64, 5mhmf 16665 . . . 4  |-  ( F  e.  ( S MndHom  U
)  ->  F :
( Base `  S ) --> ( Base `  U )
)
7 resmhm2.u . . . . . 6  |-  U  =  ( Ts  X )
87submbas 16680 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
9 eqid 2471 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
109submss 16675 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  C_  ( Base `  T ) )
118, 10eqsstr3d 3453 . . . 4  |-  ( X  e.  (SubMnd `  T
)  ->  ( Base `  U )  C_  ( Base `  T ) )
12 fss 5749 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  U )  /\  ( Base `  U )  C_  ( Base `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
136, 11, 12syl2an 485 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
14 eqid 2471 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
15 eqid 2471 . . . . . . . 8  |-  ( +g  `  U )  =  ( +g  `  U )
164, 14, 15mhmlin 16667 . . . . . . 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 1232 . . . . . 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 729 . . . . 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 2471 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
207, 19ressplusg 15317 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2120ad2antlr 741 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( +g  `  T )  =  ( +g  `  U
) )
2221oveqd 6325 . . . . 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 2508 . . . 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 2814 . . 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 2471 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
26 eqid 2471 . . . . . 6  |-  ( 0g
`  U )  =  ( 0g `  U
)
2725, 26mhm0 16668 . . . . 5  |-  ( F  e.  ( S MndHom  U
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
2827adantr 472 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
29 eqid 2471 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
307, 29subm0 16681 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3130adantl 473 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3228, 31eqtr4d 2508 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3313, 24, 323jca 1210 . 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 16662 . 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 677 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   0gc0g 15416   Mndcmnd 16613   MndHom cmhm 16658  SubMndcsubmnd 16659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661
This theorem is referenced by:  resmhm2b  16686  resghm2  16978  zrhpsgnmhm  19229  lgseisenlem4  24359
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