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

Theorem pwsplusgval 13405
Description: Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
Hypotheses
Ref Expression
pwsplusgval.y  |-  Y  =  ( R  ^s  I )
pwsplusgval.b  |-  B  =  ( Base `  Y
)
pwsplusgval.r  |-  ( ph  ->  R  e.  V )
pwsplusgval.i  |-  ( ph  ->  I  e.  W )
pwsplusgval.f  |-  ( ph  ->  F  e.  B )
pwsplusgval.g  |-  ( ph  ->  G  e.  B )
pwsplusgval.a  |-  .+  =  ( +g  `  R )
pwsplusgval.p  |-  .+b  =  ( +g  `  Y )
Assertion
Ref Expression
pwsplusgval  |-  ( ph  ->  ( F  .+b  G
)  =  ( F  o F  .+  G
) )

Proof of Theorem pwsplusgval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 eqid 2296 . . . 4  |-  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( Base `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
3 fvex 5555 . . . . 5  |-  (Scalar `  R )  e.  _V
43a1i 10 . . . 4  |-  ( ph  ->  (Scalar `  R )  e.  _V )
5 pwsplusgval.i . . . 4  |-  ( ph  ->  I  e.  W )
6 pwsplusgval.r . . . . 5  |-  ( ph  ->  R  e.  V )
7 fnconstg 5445 . . . . 5  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
86, 7syl 15 . . . 4  |-  ( ph  ->  ( I  X.  { R } )  Fn  I
)
9 pwsplusgval.f . . . . 5  |-  ( ph  ->  F  e.  B )
10 pwsplusgval.b . . . . . 6  |-  B  =  ( Base `  Y
)
11 pwsplusgval.y . . . . . . . . 9  |-  Y  =  ( R  ^s  I )
12 eqid 2296 . . . . . . . . 9  |-  (Scalar `  R )  =  (Scalar `  R )
1311, 12pwsval 13401 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
146, 5, 13syl2anc 642 . . . . . . 7  |-  ( ph  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
1610, 15syl5eq 2340 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
179, 16eleqtrd 2372 . . . 4  |-  ( ph  ->  F  e.  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
18 pwsplusgval.g . . . . 5  |-  ( ph  ->  G  e.  B )
1918, 16eleqtrd 2372 . . . 4  |-  ( ph  ->  G  e.  ( Base `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
20 eqid 2296 . . . 4  |-  ( +g  `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) )  =  ( +g  `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
211, 2, 4, 5, 8, 17, 19, 20prdsplusgval 13388 . . 3  |-  ( ph  ->  ( F ( +g  `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) G )  =  ( x  e.  I  |->  ( ( F `  x
) ( +g  `  (
( I  X.  { R } ) `  x
) ) ( G `
 x ) ) ) )
22 fvconst2g 5743 . . . . . . . 8  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
236, 22sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
2423fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( +g  `  ( ( I  X.  { R }
) `  x )
)  =  ( +g  `  R ) )
25 pwsplusgval.a . . . . . 6  |-  .+  =  ( +g  `  R )
2624, 25syl6eqr 2346 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( +g  `  ( ( I  X.  { R }
) `  x )
)  =  .+  )
2726oveqd 5891 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
) ( +g  `  (
( I  X.  { R } ) `  x
) ) ( G `
 x ) )  =  ( ( F `
 x )  .+  ( G `  x ) ) )
2827mpteq2dva 4122 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( ( I  X.  { R } ) `  x ) ) ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
)  .+  ( G `  x ) ) ) )
2921, 28eqtrd 2328 . 2  |-  ( ph  ->  ( F ( +g  `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) G )  =  ( x  e.  I  |->  ( ( F `  x
)  .+  ( G `  x ) ) ) )
30 pwsplusgval.p . . . 4  |-  .+b  =  ( +g  `  Y )
3114fveq2d 5545 . . . 4  |-  ( ph  ->  ( +g  `  Y
)  =  ( +g  `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
3230, 31syl5eq 2340 . . 3  |-  ( ph  -> 
.+b  =  ( +g  `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )
3332oveqd 5891 . 2  |-  ( ph  ->  ( F  .+b  G
)  =  ( F ( +g  `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) ) G ) )
34 fvex 5555 . . . 4  |-  ( F `
 x )  e. 
_V
3534a1i 10 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
36 fvex 5555 . . . 4  |-  ( G `
 x )  e. 
_V
3736a1i 10 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  _V )
38 eqid 2296 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3911, 38, 10, 6, 5, 9pwselbas 13404 . . . 4  |-  ( ph  ->  F : I --> ( Base `  R ) )
4039feqmptd 5591 . . 3  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
4111, 38, 10, 6, 5, 18pwselbas 13404 . . . 4  |-  ( ph  ->  G : I --> ( Base `  R ) )
4241feqmptd 5591 . . 3  |-  ( ph  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
435, 35, 37, 40, 42offval2 6111 . 2  |-  ( ph  ->  ( F  o F 
.+  G )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( G `  x )
) ) )
4429, 33, 433eqtr4d 2338 1  |-  ( ph  ->  ( F  .+b  G
)  =  ( F  o F  .+  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   X_scprds 13362    ^s cpws 13363
This theorem is referenced by:  pwsdiagmhm  14461  pwsco1mhm  14462  pwsco2mhm  14463  pwssub  14624  evl1addd  19433  mpfaddcl  19442  mpfind  19444  pf1addcl  19452  ply1rem  19565  pwssplit2  27292  frlmplusgval  27332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366
  Copyright terms: Public domain W3C validator