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Theorem frlmip 18576
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 elex 3122 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 eqid 2467 . . . . . . . 8  |-  ( R freeLMod  I )  =  ( R freeLMod  I )
4 eqid 2467 . . . . . . . 8  |-  ( Base `  ( R freeLMod  I )
)  =  ( Base `  ( R freeLMod  I )
)
53, 4frlmpws 18548 . . . . . . 7  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
62, 5sylan 471 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
76ancoms 453 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
8 fvex 5874 . . . . . . . . . 10  |-  ( (subringAlg  `  R ) `  B
)  e.  _V
9 frlmphl.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  R
)
109fveq2i 5867 . . . . . . . . . . . . . 14  |-  ( (subringAlg  `  R ) `  B
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
11 rlmval 17620 . . . . . . . . . . . . . 14  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
1210, 11eqtr4i 2499 . . . . . . . . . . . . 13  |-  ( (subringAlg  `  R ) `  B
)  =  (ringLMod `  R
)
1312eqcomi 2480 . . . . . . . . . . . 12  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  B
)
1413oveq1i 6292 . . . . . . . . . . 11  |-  ( (ringLMod `  R )  ^s  I )  =  ( ( (subringAlg  `  R ) `  B
)  ^s  I )
15 eqid 2467 . . . . . . . . . . 11  |-  (Scalar `  ( (subringAlg  `  R ) `  B ) )  =  (Scalar `  ( (subringAlg  `  R ) `  B
) )
1614, 15pwsval 14737 . . . . . . . . . 10  |-  ( ( ( (subringAlg  `  R ) `
 B )  e. 
_V  /\  I  e.  W )  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  ( (subringAlg  `  R ) `
 B ) )
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
178, 16mpan 670 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  ( (subringAlg  `  R ) `
 B ) )
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
1817adantr 465 . . . . . . . 8  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
199ressid 14546 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  R )
20 eqidd 2468 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
219eqimssi 3558 . . . . . . . . . . . . 13  |-  B  C_  ( Base `  R )
2221a1i 11 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  B  C_  ( Base `  R
) )
2320, 22srasca 17610 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  (Scalar `  ( (subringAlg  `  R ) `
 B ) ) )
2419, 23eqtr3d 2510 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  ( (subringAlg  `  R ) `  B
) ) )
2524oveq1d 6297 . . . . . . . . 9  |-  ( R  e.  V  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2625adantl 466 . . . . . . . 8  |-  ( ( 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 ) } ) ) )
2718, 26eqtr4d 2511 . . . . . . 7  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2827eqcomd 2475 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (ringLMod `  R )  ^s  I ) )
29 eqid 2467 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  Y )
301fveq2i 5867 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3129, 30eqtri 2496 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3231a1i 11 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  Y
)  =  ( Base `  ( R freeLMod  I )
) )
3328, 32oveq12d 6300 . . . . 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 ) ) ) )
347, 33eqtr4d 2511 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
351, 34syl5eq 2520 . . 3  |-  ( ( I  e.  W  /\  R  e.  V )  ->  Y  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
3635fveq2d 5868 . 2  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  Y
)  =  ( .i
`  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) ) )
37 fvex 5874 . . . 4  |-  ( Base `  Y )  e.  _V
38 eqid 2467 . . . . 5  |-  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) )  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) )
39 eqid 2467 . . . . 5  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
4038, 39ressip 14631 . . . 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
) ) ) )
4137, 40ax-mp 5 . . 3  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
42 eqid 2467 . . . . 5  |-  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )
43 simpr 461 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  R  e.  V )
44 snex 4688 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  e.  _V
45 xpexg 6709 . . . . . . 7  |-  ( ( I  e.  W  /\  { ( (subringAlg  `  R ) `
 B ) }  e.  _V )  -> 
( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
4644, 45mpan2 671 . . . . . 6  |-  ( I  e.  W  ->  (
I  X.  { ( (subringAlg  `  R ) `  B ) } )  e.  _V )
4746adantr 465 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
48 eqid 2467 . . . . 5  |-  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
49 snnzb 4092 . . . . . . . 8  |-  ( ( (subringAlg  `  R ) `  B )  e.  _V  <->  { ( (subringAlg  `  R ) `
 B ) }  =/=  (/) )
508, 49mpbi 208 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  =/=  (/)
51 dmxp 5219 . . . . . . 7  |-  ( { ( (subringAlg  `  R ) `
 B ) }  =/=  (/)  ->  dom  ( I  X.  { ( (subringAlg  `  R ) `  B
) } )  =  I )
5250, 51ax-mp 5 . . . . . 6  |-  dom  (
I  X.  { ( (subringAlg  `  R ) `  B ) } )  =  I
5352a1i 11 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  dom  ( I  X.  { ( (subringAlg  `  R
) `  B ) } )  =  I )
5442, 43, 47, 48, 53, 39prdsip 14712 . . . 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 ) ) ) ) ) )
5542, 43, 47, 48, 53prdsbas 14708 . . . . . 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 ) ) )
56 eqidd 2468 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
57 ssid 3523 . . . . . . . . . . . . 13  |-  B  C_  B
5857, 9sseqtri 3536 . . . . . . . . . . . 12  |-  B  C_  ( Base `  R )
5958rgenw 2825 . . . . . . . . . . 11  |-  A. x  e.  I  B  C_  ( Base `  R )
6059rspec 2832 . . . . . . . . . 10  |-  ( x  e.  I  ->  B  C_  ( Base `  R
) )
6156, 60srabase 17607 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  R )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
629a1i 11 . . . . . . . . 9  |-  ( x  e.  I  ->  B  =  ( Base `  R
) )
63 fvconst2g 6112 . . . . . . . . . . 11  |-  ( ( ( (subringAlg  `  R ) `
 B )  e. 
_V  /\  x  e.  I )  ->  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x )  =  ( (subringAlg  `  R
) `  B )
)
648, 63mpan 670 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x )  =  ( (subringAlg  `  R
) `  B )
)
6564fveq2d 5868 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
6661, 62, 653eqtr4rd 2519 . . . . . . . 8  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
6766adantl 466 . . . . . . 7  |-  ( ( ( I  e.  W  /\  R  e.  V
)  /\  x  e.  I )  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
6867ixpeq2dva 7481 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  -> 
X_ x  e.  I 
( Base `  ( (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) `
 x ) )  =  X_ x  e.  I  B )
69 fvex 5874 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
709, 69eqeltri 2551 . . . . . . . 8  |-  B  e. 
_V
71 ixpconstg 7475 . . . . . . . 8  |-  ( ( I  e.  W  /\  B  e.  _V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
7270, 71mpan2 671 . . . . . . 7  |-  ( I  e.  W  ->  X_ x  e.  I  B  =  ( B  ^m  I ) )
7372adantr 465 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
7455, 68, 733eqtrd 2512 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( B  ^m  I
) )
75 frlmphl.t . . . . . . . . . 10  |-  .x.  =  ( .r `  R )
7664, 60sraip 17612 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .r `  R )  =  ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) )
7775, 76syl5req 2521 . . . . . . . . 9  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  = 
.x.  )
7877oveqd 6299 . . . . . . . 8  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  `  R
) `  B ) } ) `  x
) ) ( g `
 x ) )  =  ( ( f `
 x )  .x.  ( g `  x
) ) )
7978mpteq2ia 4529 . . . . . . 7  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `
 x )  .x.  ( g `  x
) ) )
8079oveq2i 6293 . . . . . 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 )
) ) )
8180a1i 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 )
) ) ) )
8274, 74, 81mpt2eq123dv 6341 . . . 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 ) ) ) ) ) )
8354, 82eqtrd 2508 . . 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 )
) ) ) ) )
8441, 83syl5eqr 2522 . 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 )
) ) ) ) )
8536, 84eqtr2d 2509 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    ^m cmap 7417   X_cixp 7466   Basecbs 14486   ↾s cress 14487   .rcmulr 14552  Scalarcsca 14554   .icip 14556    gsumg cgsu 14692   X_scprds 14697    ^s cpws 14698  subringAlg csra 17597  ringLModcrglmod 17598   freeLMod cfrlm 18544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-prds 14699  df-pws 14701  df-sra 17601  df-rgmod 17602  df-dsmm 18530  df-frlm 18545
This theorem is referenced by:  frlmipval  18577  frlmphl  18579
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