MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixpsnbasval Structured version   Unicode version

Theorem ixpsnbasval 17290
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 7266 . . 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 5727 . . . . . . . . 9  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
) ) )
4 csbfv2g 5727 . . . . . . . . . . 11  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
)  =  ( ( { X }  X.  { (ringLMod `  R ) } ) `  [_ X  /  x ]_ x ) )
5 csbvarg 3700 . . . . . . . . . . . 12  |-  ( X  e.  W  ->  [_ X  /  x ]_ x  =  X )
65fveq2d 5695 . . . . . . . . . . 11  |-  ( X  e.  W  ->  (
( { X }  X.  { (ringLMod `  R
) } ) `  [_ X  /  x ]_ x )  =  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X ) )
74, 6eqtrd 2475 . . . . . . . . . 10  |-  ( X  e.  W  ->  [_ X  /  x ]_ ( ( { X }  X.  { (ringLMod `  R ) } ) `  x
)  =  ( ( { X }  X.  { (ringLMod `  R ) } ) `  X
) )
87fveq2d 5695 . . . . . . . . 9  |-  ( X  e.  W  ->  ( Base `  [_ X  /  x ]_ ( ( { X }  X.  {
(ringLMod `  R ) } ) `  x ) )  =  ( Base `  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X ) ) )
93, 8eqtrd 2475 . . . . . . . 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 5701 . . . . . . . . . . . . . 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 5884 . . . . . . . . . . 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 5693 . . . . . . . . 9  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X )  =  ( { <. X ,  (ringLMod `  R ) >. } `  X ) )
18 fvsng 5912 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( ( { X }  X.  { (ringLMod `  R
) } ) `  X )  =  (ringLMod `  R ) )
2120fveq2d 5695 . . . . . . 7  |-  ( ( R  e.  V  /\  X  e.  W )  ->  ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  X ) )  =  ( Base `  (ringLMod `  R ) ) )
2210, 21eqtrd 2475 . . . . . 6  |-  ( ( R  e.  V  /\  X  e.  W )  ->  [_ X  /  x ]_ ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  (ringLMod `  R ) ) )
23 rlmbas 17276 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (ringLMod `  R
) )
2422, 23syl6eqr 2493 . . . . 5  |-  ( ( R  e.  V  /\  X  e.  W )  ->  [_ X  /  x ]_ ( Base `  (
( { X }  X.  { (ringLMod `  R
) } ) `  x ) )  =  ( Base `  R
) )
2524eleq2d 2510 . . . 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 2557 . 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 2475 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 1369    e. wcel 1756   {cab 2429   _Vcvv 2972   [_csb 3288   {csn 3877   <.cop 3883    X. cxp 4838    Fn wfn 5413   ` cfv 5418   X_cixp 7263   Basecbs 14174  ringLModcrglmod 17250
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-sca 14254  df-vsca 14255  df-ip 14256  df-sra 17253  df-rgmod 17254
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator