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Theorem frlmip 18162
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 2979 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 eqid 2441 . . . . . . . 8  |-  ( R freeLMod  I )  =  ( R freeLMod  I )
4 eqid 2441 . . . . . . . 8  |-  ( Base `  ( R freeLMod  I )
)  =  ( Base `  ( R freeLMod  I )
)
53, 4frlmpws 18134 . . . . . . 7  |-  ( ( R  e.  _V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
62, 5sylan 468 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
76ancoms 450 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( (ringLMod `  R )  ^s  I )s  (
Base `  ( R freeLMod  I ) ) ) )
8 fvex 5698 . . . . . . . . . 10  |-  ( (subringAlg  `  R ) `  B
)  e.  _V
9 frlmphl.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  R
)
109fveq2i 5691 . . . . . . . . . . . . . 14  |-  ( (subringAlg  `  R ) `  B
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
11 rlmval 17250 . . . . . . . . . . . . . 14  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
1210, 11eqtr4i 2464 . . . . . . . . . . . . 13  |-  ( (subringAlg  `  R ) `  B
)  =  (ringLMod `  R
)
1312eqcomi 2445 . . . . . . . . . . . 12  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  B
)
1413oveq1i 6100 . . . . . . . . . . 11  |-  ( (ringLMod `  R )  ^s  I )  =  ( ( (subringAlg  `  R ) `  B
)  ^s  I )
15 eqid 2441 . . . . . . . . . . 11  |-  (Scalar `  ( (subringAlg  `  R ) `  B ) )  =  (Scalar `  ( (subringAlg  `  R ) `  B
) )
1614, 15pwsval 14420 . . . . . . . . . 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 665 . . . . . . . . 9  |-  ( I  e.  W  ->  (
(ringLMod `  R )  ^s  I
)  =  ( (Scalar `  ( (subringAlg  `  R ) `
 B ) )
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
1817adantr 462 . . . . . . . 8  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
199ressid 14229 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  R )
20 eqidd 2442 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
219eqimssi 3407 . . . . . . . . . . . . 13  |-  B  C_  ( Base `  R )
2221a1i 11 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  B  C_  ( Base `  R
) )
2320, 22srasca 17240 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  (Scalar `  ( (subringAlg  `  R ) `
 B ) ) )
2419, 23eqtr3d 2475 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  ( (subringAlg  `  R ) `  B
) ) )
2524oveq1d 6105 . . . . . . . . 9  |-  ( R  e.  V  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (Scalar `  ( (subringAlg  `  R ) `  B
) ) X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2625adantl 463 . . . . . . . 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 2476 . . . . . . 7  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2827eqcomd 2446 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (ringLMod `  R )  ^s  I ) )
29 eqid 2441 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  Y )
301fveq2i 5691 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3129, 30eqtri 2461 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3231a1i 11 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  Y
)  =  ( Base `  ( R freeLMod  I )
) )
3328, 32oveq12d 6108 . . . . 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 2476 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
351, 34syl5eq 2485 . . 3  |-  ( ( I  e.  W  /\  R  e.  V )  ->  Y  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
3635fveq2d 5692 . 2  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  Y
)  =  ( .i
`  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) ) )
37 fvex 5698 . . . 4  |-  ( Base `  Y )  e.  _V
38 eqid 2441 . . . . 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 2441 . . . . 5  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
4038, 39ressip 14314 . . . 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 2441 . . . . 5  |-  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )
43 simpr 458 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  R  e.  V )
44 snex 4530 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  e.  _V
45 xpexg 6506 . . . . . . 7  |-  ( ( I  e.  W  /\  { ( (subringAlg  `  R ) `
 B ) }  e.  _V )  -> 
( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
4644, 45mpan2 666 . . . . . 6  |-  ( I  e.  W  ->  (
I  X.  { ( (subringAlg  `  R ) `  B ) } )  e.  _V )
4746adantr 462 . . . . 5  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( I  X.  {
( (subringAlg  `  R ) `
 B ) } )  e.  _V )
48 eqid 2441 . . . . 5  |-  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
49 snnzb 3937 . . . . . . . 8  |-  ( ( (subringAlg  `  R ) `  B )  e.  _V  <->  { ( (subringAlg  `  R ) `
 B ) }  =/=  (/) )
508, 49mpbi 208 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  =/=  (/)
51 dmxp 5054 . . . . . . 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 14395 . . . 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 14391 . . . . . 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 2442 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
57 ssid 3372 . . . . . . . . . . . . 13  |-  B  C_  B
5857, 9sseqtri 3385 . . . . . . . . . . . 12  |-  B  C_  ( Base `  R )
5958rgenw 2781 . . . . . . . . . . 11  |-  A. x  e.  I  B  C_  ( Base `  R )
6059rspec 2778 . . . . . . . . . 10  |-  ( x  e.  I  ->  B  C_  ( Base `  R
) )
6156, 60srabase 17237 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  R )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
629a1i 11 . . . . . . . . 9  |-  ( x  e.  I  ->  B  =  ( Base `  R
) )
63 fvconst2g 5928 . . . . . . . . . . 11  |-  ( ( ( (subringAlg  `  R ) `
 B )  e. 
_V  /\  x  e.  I )  ->  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x )  =  ( (subringAlg  `  R
) `  B )
)
648, 63mpan 665 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x )  =  ( (subringAlg  `  R
) `  B )
)
6564fveq2d 5692 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
6661, 62, 653eqtr4rd 2484 . . . . . . . 8  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
6766adantl 463 . . . . . . 7  |-  ( ( ( I  e.  W  /\  R  e.  V
)  /\  x  e.  I )  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  B )
6867ixpeq2dva 7274 . . . . . 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 5698 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
709, 69eqeltri 2511 . . . . . . . 8  |-  B  e. 
_V
71 ixpconstg 7268 . . . . . . . 8  |-  ( ( I  e.  W  /\  B  e.  _V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
7270, 71mpan2 666 . . . . . . 7  |-  ( I  e.  W  ->  X_ x  e.  I  B  =  ( B  ^m  I ) )
7372adantr 462 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  -> 
X_ x  e.  I  B  =  ( B  ^m  I ) )
7455, 68, 733eqtrd 2477 . . . . 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 17242 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .r `  R )  =  ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) )
7775, 76syl5req 2486 . . . . . . . . 9  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  = 
.x.  )
7877oveqd 6107 . . . . . . . 8  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  `  R
) `  B ) } ) `  x
) ) ( g `
 x ) )  =  ( ( f `
 x )  .x.  ( g `  x
) ) )
7978mpteq2ia 4371 . . . . . . 7  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `
 x )  .x.  ( g `  x
) ) )
8079oveq2i 6101 . . . . . 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 6147 . . . 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 2473 . . 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 2487 . 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 2474 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   (/)c0 3634   {csn 3874    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7210   X_cixp 7259   Basecbs 14170   ↾s cress 14171   .rcmulr 14235  Scalarcsca 14237   .icip 14239    gsumg cgsu 14375   X_scprds 14380    ^s cpws 14381  subringAlg csra 17227  ringLModcrglmod 17228   freeLMod cfrlm 18130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-prds 14382  df-pws 14384  df-sra 17231  df-rgmod 17232  df-dsmm 18116  df-frlm 18131
This theorem is referenced by:  frlmipval  18163  frlmphl  18165
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