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Theorem frlmip 18208
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 2986 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 eqid 2443 . . . . . . . 8  |-  ( R freeLMod  I )  =  ( R freeLMod  I )
4 eqid 2443 . . . . . . . 8  |-  ( Base `  ( R freeLMod  I )
)  =  ( Base `  ( R freeLMod  I )
)
53, 4frlmpws 18180 . . . . . . 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 5706 . . . . . . . . . 10  |-  ( (subringAlg  `  R ) `  B
)  e.  _V
9 frlmphl.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  R
)
109fveq2i 5699 . . . . . . . . . . . . . 14  |-  ( (subringAlg  `  R ) `  B
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
11 rlmval 17277 . . . . . . . . . . . . . 14  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  ( Base `  R ) )
1210, 11eqtr4i 2466 . . . . . . . . . . . . 13  |-  ( (subringAlg  `  R ) `  B
)  =  (ringLMod `  R
)
1312eqcomi 2447 . . . . . . . . . . . 12  |-  (ringLMod `  R
)  =  ( (subringAlg  `  R ) `  B
)
1413oveq1i 6106 . . . . . . . . . . 11  |-  ( (ringLMod `  R )  ^s  I )  =  ( ( (subringAlg  `  R ) `  B
)  ^s  I )
15 eqid 2443 . . . . . . . . . . 11  |-  (Scalar `  ( (subringAlg  `  R ) `  B ) )  =  (Scalar `  ( (subringAlg  `  R ) `  B
) )
1614, 15pwsval 14429 . . . . . . . . . 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 14238 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  R )
20 eqidd 2444 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
219eqimssi 3415 . . . . . . . . . . . . 13  |-  B  C_  ( Base `  R )
2221a1i 11 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  B  C_  ( Base `  R
) )
2320, 22srasca 17267 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( Rs  B )  =  (Scalar `  ( (subringAlg  `  R ) `
 B ) ) )
2419, 23eqtr3d 2477 . . . . . . . . . 10  |-  ( R  e.  V  ->  R  =  (Scalar `  ( (subringAlg  `  R ) `  B
) ) )
2524oveq1d 6111 . . . . . . . . 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 2478 . . . . . . 7  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( (ringLMod `  R
)  ^s  I )  =  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
2827eqcomd 2448 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )  =  ( (ringLMod `  R )  ^s  I ) )
29 eqid 2443 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  Y )
301fveq2i 5699 . . . . . . . 8  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3129, 30eqtri 2463 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  ( R freeLMod  I ) )
3231a1i 11 . . . . . 6  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( Base `  Y
)  =  ( Base `  ( R freeLMod  I )
) )
3328, 32oveq12d 6114 . . . . 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 2478 . . . 4  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( R freeLMod  I )  =  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
351, 34syl5eq 2487 . . 3  |-  ( ( I  e.  W  /\  R  e.  V )  ->  Y  =  ( ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) )
3635fveq2d 5700 . 2  |-  ( ( I  e.  W  /\  R  e.  V )  ->  ( .i `  Y
)  =  ( .i
`  ( ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) )s  ( Base `  Y
) ) ) )
37 fvex 5706 . . . 4  |-  ( Base `  Y )  e.  _V
38 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
) )
39 eqid 2443 . . . . 5  |-  ( .i
`  ( R X_s (
I  X.  { ( (subringAlg  `  R ) `  B ) } ) ) )  =  ( .i `  ( R
X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
4038, 39ressip 14323 . . . 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 2443 . . . . 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 4538 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  e.  _V
45 xpexg 6512 . . . . . . 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 2443 . . . . 5  |-  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )  =  ( Base `  ( R X_s ( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) ) )
49 snnzb 3945 . . . . . . . 8  |-  ( ( (subringAlg  `  R ) `  B )  e.  _V  <->  { ( (subringAlg  `  R ) `
 B ) }  =/=  (/) )
508, 49mpbi 208 . . . . . . 7  |-  { ( (subringAlg  `  R ) `  B ) }  =/=  (/)
51 dmxp 5063 . . . . . . 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 14404 . . . 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 14400 . . . . . 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 2444 . . . . . . . . . 10  |-  ( x  e.  I  ->  (
(subringAlg  `  R ) `  B )  =  ( (subringAlg  `  R ) `  B ) )
57 ssid 3380 . . . . . . . . . . . . 13  |-  B  C_  B
5857, 9sseqtri 3393 . . . . . . . . . . . 12  |-  B  C_  ( Base `  R )
5958rgenw 2788 . . . . . . . . . . 11  |-  A. x  e.  I  B  C_  ( Base `  R )
6059rspec 2785 . . . . . . . . . 10  |-  ( x  e.  I  ->  B  C_  ( Base `  R
) )
6156, 60srabase 17264 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  R )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
629a1i 11 . . . . . . . . 9  |-  ( x  e.  I  ->  B  =  ( Base `  R
) )
63 fvconst2g 5936 . . . . . . . . . . 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 5700 . . . . . . . . 9  |-  ( x  e.  I  ->  ( Base `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  =  ( Base `  (
(subringAlg  `  R ) `  B ) ) )
6661, 62, 653eqtr4rd 2486 . . . . . . . 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 7283 . . . . . 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 5706 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
709, 69eqeltri 2513 . . . . . . . 8  |-  B  e. 
_V
71 ixpconstg 7277 . . . . . . . 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 2479 . . . . 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 17269 . . . . . . . . . 10  |-  ( x  e.  I  ->  ( .r `  R )  =  ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) )
7775, 76syl5req 2488 . . . . . . . . 9  |-  ( x  e.  I  ->  ( .i `  ( ( I  X.  { ( (subringAlg  `  R ) `  B
) } ) `  x ) )  = 
.x.  )
7877oveqd 6113 . . . . . . . 8  |-  ( x  e.  I  ->  (
( f `  x
) ( .i `  ( ( I  X.  { ( (subringAlg  `  R
) `  B ) } ) `  x
) ) ( g `
 x ) )  =  ( ( f `
 x )  .x.  ( g `  x
) ) )
7978mpteq2ia 4379 . . . . . . 7  |-  ( x  e.  I  |->  ( ( f `  x ) ( .i `  (
( I  X.  {
( (subringAlg  `  R ) `
 B ) } ) `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `
 x )  .x.  ( g `  x
) ) )
8079oveq2i 6107 . . . . . 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 6153 . . . 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 2475 . . 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 2489 . 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 2476 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 1369    e. wcel 1756    =/= wne 2611   _Vcvv 2977    C_ wss 3333   (/)c0 3642   {csn 3882    e. cmpt 4355    X. cxp 4843   dom cdm 4845   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    ^m cmap 7219   X_cixp 7268   Basecbs 14179   ↾s cress 14180   .rcmulr 14244  Scalarcsca 14246   .icip 14248    gsumg cgsu 14384   X_scprds 14389    ^s cpws 14390  subringAlg csra 17254  ringLModcrglmod 17255   freeLMod cfrlm 18176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-prds 14391  df-pws 14393  df-sra 17258  df-rgmod 17259  df-dsmm 18162  df-frlm 18177
This theorem is referenced by:  frlmipval  18209  frlmphl  18211
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