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Theorem ixpsnbasval 17726
Description: The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
Assertion
Ref Expression
ixpsnbasval  |-  ( ( R  e.  V  /\  X  e.  W )  -> 
X_ x  e.  { X }  ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  (
Base `  R )
) } )
Distinct variable groups:    R, f, x    f, V    f, W    f, X, x
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem ixpsnbasval
StepHypRef Expression
1 ixpsnval 7484 . . 3  |-  ( X  e.  W  ->  X_ x  e.  { X }  ( Base `  ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) )  =  { f  |  ( f  Fn 
{ X }  /\  ( f `  X
)  e.  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) ) ) } )
21adantl 466 . 2  |-  ( ( R  e.  V  /\  X  e.  W )  -> 
X_ x  e.  { X }  ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) ) ) } )
3 csbfv2g 5909 . . . . . . . . 9  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
) ) )
4 csbfv2g 5909 . . . . . . . . . . 11  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
)  =  ( ( { X }  X.  { (ringLMod `  R ) } ) `  [_ X  /  x ]_ x ) )
5 csbvarg 3853 . . . . . . . . . . . 12  |-  ( X  e.  W  ->  [_ X  /  x ]_ x  =  X )
65fveq2d 5876 . . . . . . . . . . 11  |-  ( X  e.  W  ->  (
( { X }  X.  { (ringLMod `  R
) } ) `  [_ X  /  x ]_ x )  =  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X ) )
74, 6eqtrd 2508 . . . . . . . . . 10  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
)  =  ( ( { X }  X.  { (ringLMod `  R ) } ) `  X
) )
87fveq2d 5876 . . . . . . . . 9  |-  ( X  e.  W  ->  ( Base `  [_ X  /  x ]_ ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) )  =  ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X ) ) )
93, 8eqtrd 2508 . . . . . . . 8  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  X ) ) )
109adantl 466 . . . . . . 7  |-  ( ( R  e.  V  /\  X  e.  W )  ->  [_ X  /  x ]_ ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  X ) ) )
11 fvex 5882 . . . . . . . . . . . . . 14  |-  (ringLMod `  R
)  e.  _V
1211a1i 11 . . . . . . . . . . . . 13  |-  ( R  e.  V  ->  (ringLMod `  R )  e.  _V )
1312anim1i 568 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( (ringLMod `  R
)  e.  _V  /\  X  e.  W )
)
1413ancomd 451 . . . . . . . . . . 11  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( X  e.  W  /\  (ringLMod `  R )  e.  _V ) )
15 xpsng 6073 . . . . . . . . . . 11  |-  ( ( X  e.  W  /\  (ringLMod `  R )  e. 
_V )  ->  ( { X }  X.  {
(ringLMod `  R ) } )  =  { <. X ,  (ringLMod `  R
) >. } )
1614, 15syl 16 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( { X }  X.  { (ringLMod `  R
) } )  =  { <. X ,  (ringLMod `  R ) >. } )
1716fveq1d 5874 . . . . . . . . 9  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X )  =  ( { <. X ,  (ringLMod `  R ) >. } `  X ) )
18 fvsng 6106 . . . . . . . . . 10  |-  ( ( X  e.  W  /\  (ringLMod `  R )  e. 
_V )  ->  ( { <. X ,  (ringLMod `  R ) >. } `  X )  =  (ringLMod `  R ) )
1914, 18syl 16 . . . . . . . . 9  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( { <. X , 
(ringLMod `  R ) >. } `  X )  =  (ringLMod `  R )
)
2017, 19eqtrd 2508 . . . . . . . 8  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X )  =  (ringLMod `  R ) )
2120fveq2d 5876 . . . . . . 7  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  X ) )  =  ( Base `  (ringLMod `  R ) ) )
2210, 21eqtrd 2508 . . . . . 6  |-  ( ( R  e.  V  /\  X  e.  W )  ->  [_ X  /  x ]_ ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  (ringLMod `  R ) ) )
23 rlmbas 17712 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
2422, 23syl6eqr 2526 . . . . 5  |-  ( ( R  e.  V  /\  X  e.  W )  ->  [_ X  /  x ]_ ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  R
) )
2524eleq2d 2537 . . . 4  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( f `  X )  e.  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) )  <->  ( f `  X )  e.  (
Base `  R )
) )
2625anbi2d 703 . . 3  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( f  Fn 
{ X }  /\  ( f `  X
)  e.  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) ) )  <-> 
( f  Fn  { X }  /\  (
f `  X )  e.  ( Base `  R
) ) ) )
2726abbidv 2603 . 2  |-  ( ( R  e.  V  /\  X  e.  W )  ->  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) ) ) }  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  (
Base `  R )
) } )
282, 27eqtrd 2508 1  |-  ( ( R  e.  V  /\  X  e.  W )  -> 
X_ x  e.  { X }  ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  { f  |  ( f  Fn  { X }  /\  ( f `  X )  e.  (
Base `  R )
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3118   [_csb 3440   {csn 4033   <.cop 4039    X. cxp 5003    Fn wfn 5589   ` cfv 5594   X_cixp 7481   Basecbs 14507  ringLModcrglmod 17686
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-sca 14588  df-vsca 14589  df-ip 14590  df-sra 17689  df-rgmod 17690
This theorem is referenced by: (None)
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