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Theorem pwsdiaglmhm 16053
Description: Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwsdiaglmhm.y  |-  Y  =  ( R  ^s  I )
pwsdiaglmhm.b  |-  B  =  ( Base `  R
)
pwsdiaglmhm.f  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
Assertion
Ref Expression
pwsdiaglmhm  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  F  e.  ( R LMHom  Y ) )
Distinct variable groups:    x, Y    x, R    x, I    x, B    x, W
Allowed substitution hint:    F( x)

Proof of Theorem pwsdiaglmhm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsdiaglmhm.b . 2  |-  B  =  ( Base `  R
)
2 eqid 2380 . 2  |-  ( .s
`  R )  =  ( .s `  R
)
3 eqid 2380 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
4 eqid 2380 . 2  |-  (Scalar `  R )  =  (Scalar `  R )
5 eqid 2380 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
6 eqid 2380 . 2  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 simpl 444 . 2  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  R  e.  LMod )
8 pwsdiaglmhm.y . . 3  |-  Y  =  ( R  ^s  I )
98pwslmod 15966 . 2  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  Y  e.  LMod )
108, 4pwssca 13638 . . 3  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  (Scalar `  R )  =  (Scalar `  Y ) )
1110eqcomd 2385 . 2  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  (Scalar `  Y )  =  (Scalar `  R ) )
12 lmodgrp 15877 . . 3  |-  ( R  e.  LMod  ->  R  e. 
Grp )
13 pwsdiaglmhm.f . . . 4  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
148, 1, 13pwsdiagghm 14953 . . 3  |-  ( ( R  e.  Grp  /\  I  e.  W )  ->  F  e.  ( R 
GrpHom  Y ) )
1512, 14sylan 458 . 2  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  F  e.  ( R  GrpHom  Y ) )
16 simplr 732 . . . 4  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  I  e.  W )
171, 4, 2, 6lmodvscl 15887 . . . . . 6  |-  ( ( R  e.  LMod  /\  a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )  ->  ( a ( .s
`  R ) b )  e.  B )
18173expb 1154 . . . . 5  |-  ( ( R  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  R )
)  /\  b  e.  B ) )  -> 
( a ( .s
`  R ) b )  e.  B )
1918adantlr 696 . . . 4  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( a
( .s `  R
) b )  e.  B )
2013fvdiagfn 6987 . . . 4  |-  ( ( I  e.  W  /\  ( a ( .s
`  R ) b )  e.  B )  ->  ( F `  ( a ( .s
`  R ) b ) )  =  ( I  X.  { ( a ( .s `  R ) b ) } ) )
2116, 19, 20syl2anc 643 . . 3  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( F `  ( a ( .s
`  R ) b ) )  =  ( I  X.  { ( a ( .s `  R ) b ) } ) )
2213fvdiagfn 6987 . . . . . 6  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2322ad2ant2l 727 . . . . 5  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( F `  b )  =  ( I  X.  { b } ) )
2423oveq2d 6029 . . . 4  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( a
( .s `  Y
) ( F `  b ) )  =  ( a ( .s
`  Y ) ( I  X.  { b } ) ) )
25 eqid 2380 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
26 simpll 731 . . . . 5  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  R  e.  LMod )
27 simprl 733 . . . . 5  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  a  e.  ( Base `  (Scalar `  R
) ) )
288, 1, 25pwsdiagel 13639 . . . . . 6  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  b  e.  B
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
2928adantrl 697 . . . . 5  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
308, 25, 2, 3, 4, 6, 26, 16, 27, 29pwsvscafval 13636 . . . 4  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( a
( .s `  Y
) ( I  X.  { b } ) )  =  ( ( I  X.  { a } )  o F ( .s `  R
) ( I  X.  { b } ) ) )
31 id 20 . . . . . 6  |-  ( I  e.  W  ->  I  e.  W )
32 vex 2895 . . . . . . 7  |-  a  e. 
_V
3332a1i 11 . . . . . 6  |-  ( I  e.  W  ->  a  e.  _V )
34 vex 2895 . . . . . . 7  |-  b  e. 
_V
3534a1i 11 . . . . . 6  |-  ( I  e.  W  ->  b  e.  _V )
3631, 33, 35ofc12 6261 . . . . 5  |-  ( I  e.  W  ->  (
( I  X.  {
a } )  o F ( .s `  R ) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( .s `  R ) b ) } ) )
3736ad2antlr 708 . . . 4  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( (
I  X.  { a } )  o F ( .s `  R
) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( .s `  R
) b ) } ) )
3824, 30, 373eqtrd 2416 . . 3  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( a
( .s `  Y
) ( F `  b ) )  =  ( I  X.  {
( a ( .s
`  R ) b ) } ) )
3921, 38eqtr4d 2415 . 2  |-  ( ( ( R  e.  LMod  /\  I  e.  W )  /\  ( a  e.  ( Base `  (Scalar `  R ) )  /\  b  e.  B )
)  ->  ( F `  ( a ( .s
`  R ) b ) )  =  ( a ( .s `  Y ) ( F `
 b ) ) )
401, 2, 3, 4, 5, 6, 7, 9, 11, 15, 39islmhmd 16035 1  |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  F  e.  ( R LMHom  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750    e. cmpt 4200    X. cxp 4809   ` cfv 5387  (class class class)co 6013    o Fcof 6235   Basecbs 13389  Scalarcsca 13452   .scvsca 13453    ^s cpws 13590   Grpcgrp 14605    GrpHom cghm 14923   LModclmod 15870   LMHom clmhm 16015
This theorem is referenced by:  pwslnmlem1  26856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-hom 13473  df-cco 13474  df-prds 13591  df-pws 13593  df-0g 13647  df-mnd 14610  df-mhm 14658  df-grp 14732  df-minusg 14733  df-ghm 14924  df-mgp 15569  df-rng 15583  df-ur 15585  df-lmod 15872  df-lmhm 16018
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