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Theorem asclghm 16078
Description: The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
Hypotheses
Ref Expression
asclf.a  |-  A  =  (algSc `  W )
asclf.f  |-  F  =  (Scalar `  W )
asclf.r  |-  ( ph  ->  W  e.  Ring )
asclf.l  |-  ( ph  ->  W  e.  LMod )
Assertion
Ref Expression
asclghm  |-  ( ph  ->  A  e.  ( F 
GrpHom  W ) )

Proof of Theorem asclghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2283 . 2  |-  ( Base `  W )  =  (
Base `  W )
3 eqid 2283 . 2  |-  ( +g  `  F )  =  ( +g  `  F )
4 eqid 2283 . 2  |-  ( +g  `  W )  =  ( +g  `  W )
5 asclf.l . . . 4  |-  ( ph  ->  W  e.  LMod )
6 asclf.f . . . . 5  |-  F  =  (Scalar `  W )
76lmodrng 15635 . . . 4  |-  ( W  e.  LMod  ->  F  e. 
Ring )
85, 7syl 15 . . 3  |-  ( ph  ->  F  e.  Ring )
9 rnggrp 15346 . . 3  |-  ( F  e.  Ring  ->  F  e. 
Grp )
108, 9syl 15 . 2  |-  ( ph  ->  F  e.  Grp )
11 asclf.r . . 3  |-  ( ph  ->  W  e.  Ring )
12 rnggrp 15346 . . 3  |-  ( W  e.  Ring  ->  W  e. 
Grp )
1311, 12syl 15 . 2  |-  ( ph  ->  W  e.  Grp )
14 asclf.a . . 3  |-  A  =  (algSc `  W )
1514, 6, 11, 5, 1, 2asclf 16077 . 2  |-  ( ph  ->  A : ( Base `  F ) --> ( Base `  W ) )
165adantr 451 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  W  e.  LMod )
17 simprl 732 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  x  e.  ( Base `  F
) )
18 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  y  e.  ( Base `  F
) )
19 eqid 2283 . . . . . . 7  |-  ( 1r
`  W )  =  ( 1r `  W
)
202, 19rngidcl 15361 . . . . . 6  |-  ( W  e.  Ring  ->  ( 1r
`  W )  e.  ( Base `  W
) )
2111, 20syl 15 . . . . 5  |-  ( ph  ->  ( 1r `  W
)  e.  ( Base `  W ) )
2221adantr 451 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  ( 1r `  W )  e.  ( Base `  W
) )
23 eqid 2283 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
242, 4, 6, 23, 1, 3lmodvsdir 15652 . . . 4  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
)  /\  ( 1r `  W )  e.  (
Base `  W )
) )  ->  (
( x ( +g  `  F ) y ) ( .s `  W
) ( 1r `  W ) )  =  ( ( x ( .s `  W ) ( 1r `  W
) ) ( +g  `  W ) ( y ( .s `  W
) ( 1r `  W ) ) ) )
2516, 17, 18, 22, 24syl13anc 1184 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  (
( x ( +g  `  F ) y ) ( .s `  W
) ( 1r `  W ) )  =  ( ( x ( .s `  W ) ( 1r `  W
) ) ( +g  `  W ) ( y ( .s `  W
) ( 1r `  W ) ) ) )
261, 3grpcl 14495 . . . . . 6  |-  ( ( F  e.  Grp  /\  x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) )  ->  (
x ( +g  `  F
) y )  e.  ( Base `  F
) )
27263expb 1152 . . . . 5  |-  ( ( F  e.  Grp  /\  ( x  e.  ( Base `  F )  /\  y  e.  ( Base `  F ) ) )  ->  ( x ( +g  `  F ) y )  e.  (
Base `  F )
)
2810, 27sylan 457 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  (
x ( +g  `  F
) y )  e.  ( Base `  F
) )
2914, 6, 1, 23, 19asclval 16075 . . . 4  |-  ( ( x ( +g  `  F
) y )  e.  ( Base `  F
)  ->  ( A `  ( x ( +g  `  F ) y ) )  =  ( ( x ( +g  `  F
) y ) ( .s `  W ) ( 1r `  W
) ) )
3028, 29syl 15 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  ( A `  ( x
( +g  `  F ) y ) )  =  ( ( x ( +g  `  F ) y ) ( .s
`  W ) ( 1r `  W ) ) )
3114, 6, 1, 23, 19asclval 16075 . . . . 5  |-  ( x  e.  ( Base `  F
)  ->  ( A `  x )  =  ( x ( .s `  W ) ( 1r
`  W ) ) )
3214, 6, 1, 23, 19asclval 16075 . . . . 5  |-  ( y  e.  ( Base `  F
)  ->  ( A `  y )  =  ( y ( .s `  W ) ( 1r
`  W ) ) )
3331, 32oveqan12d 5877 . . . 4  |-  ( ( x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) )  ->  (
( A `  x
) ( +g  `  W
) ( A `  y ) )  =  ( ( x ( .s `  W ) ( 1r `  W
) ) ( +g  `  W ) ( y ( .s `  W
) ( 1r `  W ) ) ) )
3433adantl 452 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  (
( A `  x
) ( +g  `  W
) ( A `  y ) )  =  ( ( x ( .s `  W ) ( 1r `  W
) ) ( +g  `  W ) ( y ( .s `  W
) ( 1r `  W ) ) ) )
3525, 30, 343eqtr4d 2325 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
) )  ->  ( A `  ( x
( +g  `  F ) y ) )  =  ( ( A `  x ) ( +g  `  W ) ( A `
 y ) ) )
361, 2, 3, 4, 10, 13, 15, 35isghmd 14692 1  |-  ( ph  ->  A  e.  ( F 
GrpHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   Grpcgrp 14362    GrpHom cghm 14680   Ringcrg 15337   1rcur 15339   LModclmod 15627  algSccascl 16052
This theorem is referenced by:  asclrhm  16081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-ascl 16055
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