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Theorem frlmlss 26631
Description: The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
frlmval.f  |-  F  =  ( R freeLMod  I )
frlmpws.b  |-  B  =  ( Base `  F
)
frlmlss.u  |-  U  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
Assertion
Ref Expression
frlmlss  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  B  e.  U )

Proof of Theorem frlmlss
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 frlmpws.b . . 3  |-  B  =  ( Base `  F
)
2 frlmval.f . . . . 5  |-  F  =  ( R freeLMod  I )
32frlmval 26628 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  =  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )
43fveq2d 5529 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  F )  =  ( Base `  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) ) )
51, 4syl5eq 2327 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  B  =  ( Base `  ( R  (+)m  ( I  X.  {
(ringLMod `  R ) } ) ) ) )
6 simpr 447 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  I  e.  W )
7 simpl 443 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  R  e.  Ring )
8 rlmlmod 15957 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
98adantr 451 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (ringLMod `  R )  e.  LMod )
10 fconst6g 5430 . . . . 5  |-  ( (ringLMod `  R )  e.  LMod  -> 
( I  X.  {
(ringLMod `  R ) } ) : I --> LMod )
119, 10syl 15 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
I  X.  { (ringLMod `  R ) } ) : I --> LMod )
12 fvex 5539 . . . . . . . 8  |-  (ringLMod `  R
)  e.  _V
1312fvconst2 5729 . . . . . . 7  |-  ( i  e.  I  ->  (
( I  X.  {
(ringLMod `  R ) } ) `  i )  =  (ringLMod `  R
) )
1413adantl 452 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  i  e.  I
)  ->  ( (
I  X.  { (ringLMod `  R ) } ) `
 i )  =  (ringLMod `  R )
)
1514fveq2d 5529 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  i  e.  I
)  ->  (Scalar `  (
( I  X.  {
(ringLMod `  R ) } ) `  i ) )  =  (Scalar `  (ringLMod `  R ) ) )
16 rlmsca 15952 . . . . . 6  |-  ( R  e.  Ring  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
1716ad2antrr 706 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  i  e.  I
)  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
1815, 17eqtr4d 2318 . . . 4  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  i  e.  I
)  ->  (Scalar `  (
( I  X.  {
(ringLMod `  R ) } ) `  i ) )  =  R )
19 eqid 2283 . . . 4  |-  ( R
X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )
20 eqid 2283 . . . 4  |-  ( LSubSp `  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )  =  ( LSubSp `  ( R X_s ( I  X.  { (ringLMod `  R ) } ) ) )
21 eqid 2283 . . . 4  |-  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) )  =  ( Base `  ( R  (+)m  ( I  X.  { (ringLMod `  R
) } ) ) )
226, 7, 11, 18, 19, 20, 21dsmmlss 26622 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  (
LSubSp `  ( R X_s (
I  X.  { (ringLMod `  R ) } ) ) ) )
23 eqid 2283 . . . . . . . . 9  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
24 eqid 2283 . . . . . . . . 9  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
2523, 24pwsval 13385 . . . . . . . 8  |-  ( ( (ringLMod `  R )  e.  _V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
2612, 25mpan 651 . . . . . . 7  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
2726adantl 452 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  (ringLMod `  R )
) X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
2816eqcomd 2288 . . . . . . . 8  |-  ( R  e.  Ring  ->  (Scalar `  (ringLMod `  R ) )  =  R )
2928adantr 451 . . . . . . 7  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (Scalar `  (ringLMod `  R )
)  =  R )
3029oveq1d 5873 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
(Scalar `  (ringLMod `  R
) ) X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) ) )
3127, 30eqtr2d 2316 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( R X_s ( I  X.  {
(ringLMod `  R ) } ) )  =  ( (ringLMod `  R )  ^s  I ) )
3231fveq2d 5529 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( LSubSp `
 ( R X_s (
I  X.  { (ringLMod `  R ) } ) ) )  =  (
LSubSp `  ( (ringLMod `  R
)  ^s  I ) ) )
33 frlmlss.u . . . 4  |-  U  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I ) )
3432, 33syl6eqr 2333 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( LSubSp `
 ( R X_s (
I  X.  { (ringLMod `  R ) } ) ) )  =  U )
3522, 34eleqtrd 2359 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  ( R  (+)m  (
I  X.  { (ringLMod `  R ) } ) ) )  e.  U
)
365, 35eqeltrd 2357 1  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   X_scprds 13346    ^s cpws 13347   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689  ringLModcrglmod 15922    (+)m cdsmm 26609   freeLMod cfrlm 26624
This theorem is referenced by:  frlm0  26634  frlmgsum  26644  frlmsplit2  26655
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-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-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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-dsmm 26610  df-frlm 26626
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