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Theorem prdsplusgval 13471
Description: Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
Hypotheses
Ref Expression
prdsbasmpt.y  |-  Y  =  ( S X_s R )
prdsbasmpt.b  |-  B  =  ( Base `  Y
)
prdsbasmpt.s  |-  ( ph  ->  S  e.  V )
prdsbasmpt.i  |-  ( ph  ->  I  e.  W )
prdsbasmpt.r  |-  ( ph  ->  R  Fn  I )
prdsplusgval.f  |-  ( ph  ->  F  e.  B )
prdsplusgval.g  |-  ( ph  ->  G  e.  B )
prdsplusgval.p  |-  .+  =  ( +g  `  Y )
Assertion
Ref Expression
prdsplusgval  |-  ( ph  ->  ( F  .+  G
)  =  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x )
) ( G `  x ) ) ) )
Distinct variable groups:    x, B    x, F    x, G    ph, x    x, I    x, V    x, R    x, S    x, W    x, Y
Allowed substitution hint:    .+ ( x)

Proof of Theorem prdsplusgval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbasmpt.y . . 3  |-  Y  =  ( S X_s R )
2 prdsbasmpt.s . . 3  |-  ( ph  ->  S  e.  V )
3 prdsbasmpt.r . . . 4  |-  ( ph  ->  R  Fn  I )
4 prdsbasmpt.i . . . 4  |-  ( ph  ->  I  e.  W )
5 fnex 5827 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  W )  ->  R  e.  _V )
63, 4, 5syl2anc 642 . . 3  |-  ( ph  ->  R  e.  _V )
7 prdsbasmpt.b . . 3  |-  B  =  ( Base `  Y
)
8 fndm 5425 . . . 4  |-  ( R  Fn  I  ->  dom  R  =  I )
93, 8syl 15 . . 3  |-  ( ph  ->  dom  R  =  I )
10 prdsplusgval.p . . 3  |-  .+  =  ( +g  `  Y )
111, 2, 6, 7, 9, 10prdsplusg 13457 . 2  |-  ( ph  ->  .+  =  ( y  e.  B ,  z  e.  B  |->  ( x  e.  I  |->  ( ( y `  x ) ( +g  `  ( R `  x )
) ( z `  x ) ) ) ) )
12 fveq1 5607 . . . . 5  |-  ( y  =  F  ->  (
y `  x )  =  ( F `  x ) )
13 fveq1 5607 . . . . 5  |-  ( z  =  G  ->  (
z `  x )  =  ( G `  x ) )
1412, 13oveqan12d 5964 . . . 4  |-  ( ( y  =  F  /\  z  =  G )  ->  ( ( y `  x ) ( +g  `  ( R `  x
) ) ( z `
 x ) )  =  ( ( F `
 x ) ( +g  `  ( R `
 x ) ) ( G `  x
) ) )
1514adantl 452 . . 3  |-  ( (
ph  /\  ( y  =  F  /\  z  =  G ) )  -> 
( ( y `  x ) ( +g  `  ( R `  x
) ) ( z `
 x ) )  =  ( ( F `
 x ) ( +g  `  ( R `
 x ) ) ( G `  x
) ) )
1615mpteq2dv 4188 . 2  |-  ( (
ph  /\  ( y  =  F  /\  z  =  G ) )  -> 
( x  e.  I  |->  ( ( y `  x ) ( +g  `  ( R `  x
) ) ( z `
 x ) ) )  =  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x )
) ( G `  x ) ) ) )
17 prdsplusgval.f . 2  |-  ( ph  ->  F  e.  B )
18 prdsplusgval.g . 2  |-  ( ph  ->  G  e.  B )
19 mptexg 5831 . . 3  |-  ( I  e.  W  ->  (
x  e.  I  |->  ( ( F `  x
) ( +g  `  ( R `  x )
) ( G `  x ) ) )  e.  _V )
204, 19syl 15 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x
) ) ( G `
 x ) ) )  e.  _V )
2111, 16, 17, 18, 20ovmpt2d 6062 1  |-  ( ph  ->  ( F  .+  G
)  =  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x )
) ( G `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    e. cmpt 4158   dom cdm 4771    Fn wfn 5332   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   X_scprds 13445
This theorem is referenced by:  prdsplusgfval  13472  pwsplusgval  13488  xpsadd  13577  prdsplusgcl  14502  prdsidlem  14503  prdsmndd  14504  prdsinvlem  14702  prdscmnd  15252  prdsrngd  15494  prdslmodd  15825  prdstmdd  17908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-hom 13329  df-cco 13330  df-prds 13447
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