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Theorem frlmip 18787
Description: The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
Hypotheses
Ref Expression
frlmphl.y  |-  Y  =  ( R freeLMod  I )
frlmphl.b  |-  B  =  ( Base `  R
)
frlmphl.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
frlmip  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( f  e.  ( B  ^m  I ) ,  g  e.  ( B  ^m  I ) 
|->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) ) )  =  ( .i `  Y ) )
Distinct variable groups:    B, f,
g, x    f, I,
g, x    R, f,
g, x    f, V, g, x    f, W, g, x
Allowed substitution hints:    .x. ( x, f, g)    Y( x, f, g)

Proof of Theorem frlmip
StepHypRef Expression
1 frlmphl.y . . . 4  |-  Y  =  ( R freeLMod  I )
2 eqid 2443 . . . . . . 7  |-  ( R freeLMod  I )  =  ( R freeLMod  I )
3 eqid 2443 . . . . . . 7  |-  ( Base `  ( R freeLMod  I )
)  =  ( Base `  ( R freeLMod  I )
)
42, 3frlmpws 18759 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
54ancoms 453 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
6 frlmphl.b . . . . . . . . . . 11  |-  B  =  ( Base `  R
)
76ressid 14674 . . . . . . . . . 10  |-  ( R  e.  V  ->  ( Rs  B )  =  R )
8 eqidd 2444 . . . . . . . . . . 11  |-  ( R  e.  V  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
96eqimssi 3543 . . . . . . . . . . . 12  |-  B  C_  ( Base `  R )
109a1i 11 . . . . . . . . . . 11  |-  ( R  e.  V  ->  B  C_  ( Base `  R
) )
118, 10srasca 17806 . . . . . . . . . 10  |-  ( R  e.  V  ->  ( Rs  B )  =  (Scalar `  ( (subringAlg  `  R ) `
 B ) ) )
127, 11eqtr3d 2486 . . . . . . . . 9  |-  ( R  e.  V  ->  R  =  (Scalar `  ( (subringAlg  `  R ) `  B
) ) )
1312oveq1d 6296 . . . . . . . 8  |-  ( R  e.  V  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
1413adantl 466 . . . . . . 7  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
15 fvex 5866 . . . . . . . . 9  |-  ( (subringAlg  `  R ) `  B
)  e.  _V
16 rlmval 17816 . . . . . . . . . . . 12  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
176fveq2i 5859 . . . . . . . . . . . 12  |-  ( (subringAlg  `  R ) `  B
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
1816, 17eqtr4i 2475 . . . . . . . . . . 11  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  B
)
1918oveq1i 6291 . . . . . . . . . 10  |-  ( (ringLMod `  R )  ^s  I )  =  ( ( (subringAlg  `  R ) `  B
)  ^s  I )
20 eqid 2443 . . . . . . . . . 10  |-  (Scalar `  ( (subringAlg  `  R ) `  B ) )  =  (Scalar `  ( (subringAlg  `  R ) `  B
) )
2119, 20pwsval 14865 . . . . . . . . 9  |-  ( ( ( (subringAlg  `  R ) `
 B )  e. 
_V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  ( (subringAlg  `  R ) `
 B ) )
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2215, 21mpan 670 . . . . . . . 8  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  ( (subringAlg  `  R ) `
 B ) )
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2322adantr 465 . . . . . . 7  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2414, 23eqtr4d 2487 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (ringLMod `  R )  ^s  I ) )
251fveq2i 5859 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
2625a1i 11 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  Y
)  =  ( Base `  ( R freeLMod  I )
) )
2724, 26oveq12d 6299 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) )s  ( Base `  Y
) )  =  ( ( (ringLMod `  R
)  ^s  I )s  ( Base `  ( R freeLMod  I ) ) ) )
285, 27eqtr4d 2487 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
291, 28syl5eq 2496 . . 3  |-  ( ( I  e.  W  /\  R  e.  V )  ->  Y  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
3029fveq2d 5860 . 2  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  Y
)  =  ( .i
`  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) ) )
31 fvex 5866 . . . 4  |-  ( Base `  Y )  e.  _V
32 eqid 2443 . . . . 5  |-  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) )  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) )
33 eqid 2443 . . . . 5  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
3432, 33ressip 14759 . . . 4  |-  ( (
Base `  Y )  e.  _V  ->  ( .i `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( .i `  (
( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) ) )
3531, 34ax-mp 5 . . 3  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
36 eqid 2443 . . . . 5  |-  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )
37 simpr 461 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  R  e.  V )
38 snex 4678 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  e.  _V
39 xpexg 6587 . . . . . . 7  |-  ( ( I  e.  W  /\  { ( (subringAlg  `  R ) `
 B ) }  e.  _V )  -> 
( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
4038, 39mpan2 671 . . . . . 6  |-  ( I  e.  W  ->  (
I  X.  { ( (subringAlg  `  R ) `  B ) } )  e.  _V )
4140adantr 465 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
42 eqid 2443 . . . . 5  |-  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
4315snnz 4133 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  =/=  (/)
44 dmxp 5211 . . . . . . 7  |-  ( { ( (subringAlg  `  R ) `
 B ) }  =/=  (/)  ->  dom  ( I  X.  { ( (subringAlg  `  R ) `  B
) } )  =  I )
4543, 44ax-mp 5 . . . . . 6  |-  dom  (
I  X.  { ( (subringAlg  `  R ) `  B ) } )  =  I
4645a1i 11 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  dom  ( I  X.  { ( (subringAlg  `  R
) `  B ) } )  =  I )
4736, 37, 41, 42, 46, 33prdsip 14840 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( f  e.  (
Base `  ( R X_s ( I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) ) ,  g  e.  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  |->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) ) ( g `  x ) ) ) ) ) )
4836, 37, 41, 42, 46prdsbas 14836 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  = 
X_ x  e.  I 
( Base `  ( (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) `
 x ) ) )
49 eqidd 2444 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
509a1i 11 . . . . . . . . . 10  |-  ( x  e.  I  ->  B  C_  ( Base `  R
) )
5149, 50srabase 17803 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  R )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
526a1i 11 . . . . . . . . 9  |-  ( x  e.  I  ->  B  =  ( Base `  R
) )
5315fvconst2 6111 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x )  =  ( (subringAlg  `  R
) `  B )
)
5453fveq2d 5860 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
5551, 52, 543eqtr4rd 2495 . . . . . . . 8  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
5655adantl 466 . . . . . . 7  |-  ( ( ( I  e.  W  /\  R  e.  V
)  /\  x  e.  I )  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
5756ixpeq2dva 7486 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  -> 
X_ x  e.  I 
( Base `  ( (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) `
 x ) )  =  X_ x  e.  I  B )
58 fvex 5866 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
596, 58eqeltri 2527 . . . . . . . 8  |-  B  e. 
_V
60 ixpconstg 7480 . . . . . . . 8  |-  ( ( I  e.  W  /\  B  e.  _V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
6159, 60mpan2 671 . . . . . . 7  |-  ( I  e.  W  ->  X_ x  e.  I  B  =  ( B  ^m  I ) )
6261adantr 465 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
6348, 57, 623eqtrd 2488 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( B  ^m  I
) )
64 frlmphl.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
6553, 50sraip 17808 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .r `  R )  =  ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) )
6664, 65syl5req 2497 . . . . . . . . 9  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  = 
.x.  )
6766oveqd 6298 . . . . . . . 8  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  `  R
) `  B ) } ) `  x
) ) ( g `
 x ) )  =  ( ( f `
 x )  .x.  ( g `  x
) ) )
6867mpteq2ia 4519 . . . . . . 7  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `
 x )  .x.  ( g `  x
) ) )
6968oveq2i 6292 . . . . . 6  |-  ( R 
gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) ) ( g `  x ) ) ) )  =  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) )
7069a1i 11 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) ) ( g `  x ) ) ) )  =  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) ) )
7163, 63, 70mpt2eq123dv 6344 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( f  e.  (
Base `  ( R X_s ( I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) ) ,  g  e.  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  |->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) ) ( g `  x ) ) ) ) )  =  ( f  e.  ( B  ^m  I
) ,  g  e.  ( B  ^m  I
)  |->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x ) 
.x.  ( g `  x ) ) ) ) ) )
7247, 71eqtrd 2484 . . 3  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( f  e.  ( B  ^m  I ) ,  g  e.  ( B  ^m  I ) 
|->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) ) ) )
7335, 72syl5eqr 2498 . 2  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  (
( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )  =  ( f  e.  ( B  ^m  I ) ,  g  e.  ( B  ^m  I ) 
|->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) ) ) )
7430, 73eqtr2d 2485 1  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( f  e.  ( B  ^m  I ) ,  g  e.  ( B  ^m  I ) 
|->  ( R  gsumg  ( x  e.  I  |->  ( ( f `  x )  .x.  (
g `  x )
) ) ) )  =  ( .i `  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    C_ wss 3461   (/)c0 3770   {csn 4014    |-> cmpt 4495    X. cxp 4987   dom cdm 4989   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    ^m cmap 7422   X_cixp 7471   Basecbs 14614   ↾s cress 14615   .rcmulr 14680  Scalarcsca 14682   .icip 14684    gsumg cgsu 14820   X_scprds 14825    ^s cpws 14826  subringAlg csra 17793  ringLModcrglmod 17794   freeLMod cfrlm 18755
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-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 10987  df-uz 11093  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-prds 14827  df-pws 14829  df-sra 17797  df-rgmod 17798  df-dsmm 18741  df-frlm 18756
This theorem is referenced by:  frlmipval  18788  frlmphl  18790
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