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Theorem frlmvscafval 27221
Description: Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
frlmvscafval.y  |-  Y  =  ( R freeLMod  I )
frlmvscafval.b  |-  B  =  ( Base `  Y
)
frlmvscafval.k  |-  K  =  ( Base `  R
)
frlmvscafval.i  |-  ( ph  ->  I  e.  W )
frlmvscafval.a  |-  ( ph  ->  A  e.  K )
frlmvscafval.x  |-  ( ph  ->  X  e.  B )
frlmvscafval.v  |-  .xb  =  ( .s `  Y )
frlmvscafval.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
frlmvscafval  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )

Proof of Theorem frlmvscafval
StepHypRef Expression
1 frlmvscafval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
2 frlmvscafval.y . . . . . . . 8  |-  Y  =  ( R freeLMod  I )
3 frlmvscafval.b . . . . . . . 8  |-  B  =  ( Base `  Y
)
42, 3frlmrcl 27216 . . . . . . 7  |-  ( X  e.  B  ->  R  e.  _V )
51, 4syl 16 . . . . . 6  |-  ( ph  ->  R  e.  _V )
6 frlmvscafval.i . . . . . 6  |-  ( ph  ->  I  e.  W )
72, 3frlmpws 27209 . . . . . 6  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
85, 6, 7syl2anc 644 . . . . 5  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
98fveq2d 5735 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
10 frlmvscafval.v . . . 4  |-  .xb  =  ( .s `  Y )
11 fvex 5745 . . . . . 6  |-  ( Base `  Y )  e.  _V
123, 11eqeltri 2508 . . . . 5  |-  B  e. 
_V
13 eqid 2438 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
14 eqid 2438 . . . . . 6  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
(ringLMod `  R )  ^s  I
) )
1513, 14ressvsca 13610 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
1612, 15ax-mp 5 . . . 4  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
( (ringLMod `  R )  ^s  I )s  B ) )
179, 10, 163eqtr4g 2495 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) )
1817oveqd 6101 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  (
(ringLMod `  R )  ^s  I
) ) X ) )
19 eqid 2438 . . 3  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
20 eqid 2438 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
21 frlmvscafval.t . . . 4  |-  .x.  =  ( .r `  R )
22 rlmvsca 16278 . . . 4  |-  ( .r
`  R )  =  ( .s `  (ringLMod `  R ) )
2321, 22eqtri 2458 . . 3  |-  .x.  =  ( .s `  (ringLMod `  R
) )
24 eqid 2438 . . 3  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
25 eqid 2438 . . 3  |-  ( Base `  (Scalar `  (ringLMod `  R
) ) )  =  ( Base `  (Scalar `  (ringLMod `  R )
) )
26 fvex 5745 . . . 4  |-  (ringLMod `  R
)  e.  _V
2726a1i 11 . . 3  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
28 frlmvscafval.a . . . 4  |-  ( ph  ->  A  e.  K )
29 frlmvscafval.k . . . . 5  |-  K  =  ( Base `  R
)
30 rlmsca 16276 . . . . . . 7  |-  ( R  e.  _V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
315, 30syl 16 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
3231fveq2d 5735 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3329, 32syl5eq 2482 . . . 4  |-  ( ph  ->  K  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3428, 33eleqtrd 2514 . . 3  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
358fveq2d 5735 . . . . . 6  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
363, 35syl5eq 2482 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
3713, 20ressbasss 13526 . . . . 5  |-  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) )  C_  ( Base `  ( (ringLMod `  R )  ^s  I ) )
3836, 37syl6eqss 3400 . . . 4  |-  ( ph  ->  B  C_  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
3938, 1sseldd 3351 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
4019, 20, 23, 14, 24, 25, 27, 6, 34, 39pwsvscafval 13721 . 2  |-  ( ph  ->  ( A ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) X )  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
4118, 40eqtrd 2470 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4879   ` cfv 5457  (class class class)co 6084    o Fcof 6306   Basecbs 13474   ↾s cress 13475   .rcmulr 13535  Scalarcsca 13537   .scvsca 13538    ^s cpws 13675  ringLModcrglmod 16246   freeLMod cfrlm 27203
This theorem is referenced by:  frlmvscaval  27222  uvcresum  27233  matvsca2  27469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-hom 13558  df-cco 13559  df-prds 13676  df-pws 13678  df-sra 16249  df-rgmod 16250  df-dsmm 27189  df-frlm 27205
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