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Theorem resmhm2 16193
Description: One direction of resmhm2b 16194. (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 16171 . . 3  |-  ( F  e.  ( S MndHom  U
)  ->  S  e.  Mnd )
2 submrcl 16179 . . 3  |-  ( X  e.  (SubMnd `  T
)  ->  T  e.  Mnd )
31, 2anim12i 564 . 2  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
4 eqid 2454 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
5 eqid 2454 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
64, 5mhmf 16173 . . . 4  |-  ( F  e.  ( S MndHom  U
)  ->  F :
( Base `  S ) --> ( Base `  U )
)
7 resmhm2.u . . . . . 6  |-  U  =  ( Ts  X )
87submbas 16188 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  =  ( Base `  U )
)
9 eqid 2454 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
109submss 16183 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  X  C_  ( Base `  T ) )
118, 10eqsstr3d 3524 . . . 4  |-  ( X  e.  (SubMnd `  T
)  ->  ( Base `  U )  C_  ( Base `  T ) )
12 fss 5721 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  U )  /\  ( Base `  U )  C_  ( Base `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
136, 11, 12syl2an 475 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
14 eqid 2454 . . . . . . . 8  |-  ( +g  `  S )  =  ( +g  `  S )
15 eqid 2454 . . . . . . . 8  |-  ( +g  `  U )  =  ( +g  `  U )
164, 14, 15mhmlin 16175 . . . . . . 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 1195 . . . . . 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 712 . . . . 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 2454 . . . . . . . 8  |-  ( +g  `  T )  =  ( +g  `  T )
207, 19ressplusg 14833 . . . . . . 7  |-  ( X  e.  (SubMnd `  T
)  ->  ( +g  `  T )  =  ( +g  `  U ) )
2120ad2antlr 724 . . . . . 6  |-  ( ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T
) )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( +g  `  T )  =  ( +g  `  U
) )
2221oveqd 6287 . . . . 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 2498 . . . 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 2875 . . 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 2454 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
26 eqid 2454 . . . . . 6  |-  ( 0g
`  U )  =  ( 0g `  U
)
2725, 26mhm0 16176 . . . . 5  |-  ( F  e.  ( S MndHom  U
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
2827adantr 463 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  U ) )
29 eqid 2454 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
307, 29subm0 16189 . . . . 5  |-  ( X  e.  (SubMnd `  T
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3130adantl 464 . . . 4  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( 0g `  T )  =  ( 0g `  U ) )
3228, 31eqtr4d 2498 . . 3  |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T )
)  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
3313, 24, 323jca 1174 . 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 16170 . 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 662 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   ↾s cress 14720   +g cplusg 14787   0gc0g 14932   Mndcmnd 16121   MndHom cmhm 16166  SubMndcsubmnd 16167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169
This theorem is referenced by:  resmhm2b  16194  resghm2  16486  zrhpsgnmhm  18796  lgseisenlem4  23828
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