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Theorem frlmvscafval 27145
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 27140 . . . . . . 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 27133 . . . . . 6  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
85, 6, 7syl2anc 643 . . . . 5  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
98fveq2d 5723 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
10 frlmvscafval.v . . . 4  |-  .xb  =  ( .s `  Y )
11 fvex 5733 . . . . . 6  |-  ( Base `  Y )  e.  _V
123, 11eqeltri 2505 . . . . 5  |-  B  e. 
_V
13 eqid 2435 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
14 eqid 2435 . . . . . 6  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
(ringLMod `  R )  ^s  I
) )
1513, 14ressvsca 13593 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
( (ringLMod `  R )  ^s  I )s  B ) ) )
1612, 15ax-mp 8 . . . 4  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
( (ringLMod `  R )  ^s  I )s  B ) )
179, 10, 163eqtr4g 2492 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) )
1817oveqd 6089 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  (
(ringLMod `  R )  ^s  I
) ) X ) )
19 eqid 2435 . . 3  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
20 eqid 2435 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
21 frlmvscafval.t . . . 4  |-  .x.  =  ( .r `  R )
22 rlmvsca 16261 . . . 4  |-  ( .r
`  R )  =  ( .s `  (ringLMod `  R ) )
2321, 22eqtri 2455 . . 3  |-  .x.  =  ( .s `  (ringLMod `  R
) )
24 eqid 2435 . . 3  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
25 eqid 2435 . . 3  |-  ( Base `  (Scalar `  (ringLMod `  R
) ) )  =  ( Base `  (Scalar `  (ringLMod `  R )
) )
26 fvex 5733 . . . 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 16259 . . . . . . 7  |-  ( R  e.  _V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
315, 30syl 16 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
3231fveq2d 5723 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3329, 32syl5eq 2479 . . . 4  |-  ( ph  ->  K  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3428, 33eleqtrd 2511 . . 3  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
358fveq2d 5723 . . . . . 6  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
363, 35syl5eq 2479 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
3713, 20ressbasss 13509 . . . . 5  |-  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) )  C_  ( Base `  ( (ringLMod `  R )  ^s  I ) )
3836, 37syl6eqss 3390 . . . 4  |-  ( ph  ->  B  C_  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
3938, 1sseldd 3341 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
4019, 20, 23, 14, 24, 25, 27, 6, 34, 39pwsvscafval 13704 . 2  |-  ( ph  ->  ( A ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) X )  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
4118, 40eqtrd 2467 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    X. cxp 4867   ` cfv 5445  (class class class)co 6072    o Fcof 6294   Basecbs 13457   ↾s cress 13458   .rcmulr 13518  Scalarcsca 13520   .scvsca 13521    ^s cpws 13658  ringLModcrglmod 16229   freeLMod cfrlm 27127
This theorem is referenced by:  frlmvscaval  27146  uvcresum  27157  matvsca2  27393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-hom 13541  df-cco 13542  df-prds 13659  df-pws 13661  df-sra 16232  df-rgmod 16233  df-dsmm 27113  df-frlm 27129
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