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Theorem frlmvscafval 27230
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 27225 . . . . . . 7  |-  ( X  e.  B  ->  R  e.  _V )
51, 4syl 15 . . . . . 6  |-  ( ph  ->  R  e.  _V )
6 frlmvscafval.i . . . . . 6  |-  ( ph  ->  I  e.  W )
72, 3frlmpws 27218 . . . . . 6  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
85, 6, 7syl2anc 642 . . . . 5  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
98fveq2d 5529 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
10 frlmvscafval.v . . . 4  |-  .xb  =  ( .s `  Y )
11 fvex 5539 . . . . . 6  |-  ( Base `  Y )  e.  _V
123, 11eqeltri 2353 . . . . 5  |-  B  e. 
_V
13 eqid 2283 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
14 eqid 2283 . . . . . 6  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
(ringLMod `  R )  ^s  I
) )
1513, 14ressvsca 13284 . . . . 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 2340 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) )
1817oveqd 5875 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  (
(ringLMod `  R )  ^s  I
) ) X ) )
19 eqid 2283 . . 3  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
20 eqid 2283 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
21 frlmvscafval.t . . . 4  |-  .x.  =  ( .r `  R )
22 rlmvsca 15954 . . . 4  |-  ( .r
`  R )  =  ( .s `  (ringLMod `  R ) )
2321, 22eqtri 2303 . . 3  |-  .x.  =  ( .s `  (ringLMod `  R
) )
24 eqid 2283 . . 3  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
25 eqid 2283 . . 3  |-  ( Base `  (Scalar `  (ringLMod `  R
) ) )  =  ( Base `  (Scalar `  (ringLMod `  R )
) )
26 fvex 5539 . . . 4  |-  (ringLMod `  R
)  e.  _V
2726a1i 10 . . 3  |-  ( ph  ->  (ringLMod `  R )  e.  _V )
28 frlmvscafval.a . . . 4  |-  ( ph  ->  A  e.  K )
29 frlmvscafval.k . . . . 5  |-  K  =  ( Base `  R
)
30 rlmsca 15952 . . . . . . 7  |-  ( R  e.  _V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
315, 30syl 15 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
3231fveq2d 5529 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3329, 32syl5eq 2327 . . . 4  |-  ( ph  ->  K  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3428, 33eleqtrd 2359 . . 3  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
358fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
363, 35syl5eq 2327 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
3713, 20ressbasss 13200 . . . . . 6  |-  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) )  C_  ( Base `  ( (ringLMod `  R )  ^s  I ) )
3837a1i 10 . . . . 5  |-  ( ph  ->  ( Base `  (
( (ringLMod `  R )  ^s  I )s  B ) )  C_  ( Base `  ( (ringLMod `  R )  ^s  I ) ) )
3936, 38eqsstrd 3212 . . . 4  |-  ( ph  ->  B  C_  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
4039, 1sseldd 3181 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
4119, 20, 23, 14, 24, 25, 27, 6, 34, 40pwsvscafval 13393 . 2  |-  ( ph  ->  ( A ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) X )  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
4218, 41eqtrd 2315 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   ↾s cress 13149   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212    ^s cpws 13347  ringLModcrglmod 15922   freeLMod cfrlm 27212
This theorem is referenced by:  frlmvscaval  27231  uvcresum  27242  matvsca2  27478
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-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-int 3863  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-sra 15925  df-rgmod 15926  df-dsmm 27198  df-frlm 27214
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