MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frlmvscafval Structured version   Unicode version

Theorem frlmvscafval 18776
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 } )  oF  .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 18767 . . . . . . 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 18758 . . . . . 6  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
85, 6, 7syl2anc 661 . . . . 5  |-  ( ph  ->  Y  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
98fveq2d 5860 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( ( (ringLMod `  R )  ^s  I )s  B ) ) )
10 frlmvscafval.v . . . 4  |-  .xb  =  ( .s `  Y )
11 fvex 5866 . . . . . 6  |-  ( Base `  Y )  e.  _V
123, 11eqeltri 2527 . . . . 5  |-  B  e. 
_V
13 eqid 2443 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  I )s  B )  =  ( ( (ringLMod `  R
)  ^s  I )s  B )
14 eqid 2443 . . . . . 6  |-  ( .s
`  ( (ringLMod `  R
)  ^s  I ) )  =  ( .s `  (
(ringLMod `  R )  ^s  I
) )
1513, 14ressvsca 14757 . . . . 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 2509 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) )
1817oveqd 6298 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  (
(ringLMod `  R )  ^s  I
) ) X ) )
19 eqid 2443 . . 3  |-  ( (ringLMod `  R )  ^s  I )  =  ( (ringLMod `  R
)  ^s  I )
20 eqid 2443 . . 3  |-  ( Base `  ( (ringLMod `  R
)  ^s  I ) )  =  ( Base `  (
(ringLMod `  R )  ^s  I
) )
21 frlmvscafval.t . . . 4  |-  .x.  =  ( .r `  R )
22 rlmvsca 17826 . . . 4  |-  ( .r
`  R )  =  ( .s `  (ringLMod `  R ) )
2321, 22eqtri 2472 . . 3  |-  .x.  =  ( .s `  (ringLMod `  R
) )
24 eqid 2443 . . 3  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
25 eqid 2443 . . 3  |-  ( Base `  (Scalar `  (ringLMod `  R
) ) )  =  ( Base `  (Scalar `  (ringLMod `  R )
) )
26 fvex 5866 . . . 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 17824 . . . . . . 7  |-  ( R  e.  _V  ->  R  =  (Scalar `  (ringLMod `  R
) ) )
315, 30syl 16 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
3231fveq2d 5860 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3329, 32syl5eq 2496 . . . 4  |-  ( ph  ->  K  =  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
3428, 33eleqtrd 2533 . . 3  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  (ringLMod `  R
) ) ) )
358fveq2d 5860 . . . . . 6  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
363, 35syl5eq 2496 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) ) )
3713, 20ressbasss 14670 . . . . 5  |-  ( Base `  ( ( (ringLMod `  R
)  ^s  I )s  B ) )  C_  ( Base `  ( (ringLMod `  R )  ^s  I ) )
3836, 37syl6eqss 3539 . . . 4  |-  ( ph  ->  B  C_  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
3938, 1sseldd 3490 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( (ringLMod `  R
)  ^s  I ) ) )
4019, 20, 23, 14, 24, 25, 27, 6, 34, 39pwsvscafval 14872 . 2  |-  ( ph  ->  ( A ( .s
`  ( (ringLMod `  R
)  ^s  I ) ) X )  =  ( ( I  X.  { A } )  oF  .x.  X ) )
4118, 40eqtrd 2484 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  oF  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095   {csn 4014    X. cxp 4987   ` cfv 5578  (class class class)co 6281    oFcof 6523   Basecbs 14613   ↾s cress 14614   .rcmulr 14679  Scalarcsca 14681   .scvsca 14682    ^s cpws 14825  ringLModcrglmod 17793   freeLMod cfrlm 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-hom 14702  df-cco 14703  df-prds 14826  df-pws 14828  df-sra 17796  df-rgmod 17797  df-dsmm 18740  df-frlm 18755
This theorem is referenced by:  frlmvscaval  18777  uvcresum  18801  matvsca2  18907  zlmodzxzscm  32679
  Copyright terms: Public domain W3C validator