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Theorem prdsplusgcl 14654
Description: Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Hypotheses
Ref Expression
prdsplusgcl.y  |-  Y  =  ( S X_s R )
prdsplusgcl.b  |-  B  =  ( Base `  Y
)
prdsplusgcl.p  |-  .+  =  ( +g  `  Y )
prdsplusgcl.s  |-  ( ph  ->  S  e.  V )
prdsplusgcl.i  |-  ( ph  ->  I  e.  W )
prdsplusgcl.r  |-  ( ph  ->  R : I --> Mnd )
prdsplusgcl.f  |-  ( ph  ->  F  e.  B )
prdsplusgcl.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
prdsplusgcl  |-  ( ph  ->  ( F  .+  G
)  e.  B )

Proof of Theorem prdsplusgcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prdsplusgcl.y . . 3  |-  Y  =  ( S X_s R )
2 prdsplusgcl.b . . 3  |-  B  =  ( Base `  Y
)
3 prdsplusgcl.s . . 3  |-  ( ph  ->  S  e.  V )
4 prdsplusgcl.i . . 3  |-  ( ph  ->  I  e.  W )
5 prdsplusgcl.r . . . 4  |-  ( ph  ->  R : I --> Mnd )
6 ffn 5532 . . . 4  |-  ( R : I --> Mnd  ->  R  Fn  I )
75, 6syl 16 . . 3  |-  ( ph  ->  R  Fn  I )
8 prdsplusgcl.f . . 3  |-  ( ph  ->  F  e.  B )
9 prdsplusgcl.g . . 3  |-  ( ph  ->  G  e.  B )
10 prdsplusgcl.p . . 3  |-  .+  =  ( +g  `  Y )
111, 2, 3, 4, 7, 8, 9, 10prdsplusgval 13623 . 2  |-  ( ph  ->  ( F  .+  G
)  =  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x )
) ( G `  x ) ) ) )
125ffvelrnda 5810 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  Mnd )
133adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  V )
144adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
157adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
168adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  B )
17 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
181, 2, 13, 14, 15, 16, 17prdsbasprj 13622 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( Base `  ( R `  x )
) )
199adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  B )
201, 2, 13, 14, 15, 19, 17prdsbasprj 13622 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  ( R `  x )
) )
21 eqid 2388 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
22 eqid 2388 . . . . . 6  |-  ( +g  `  ( R `  x
) )  =  ( +g  `  ( R `
 x ) )
2321, 22mndcl 14623 . . . . 5  |-  ( ( ( R `  x
)  e.  Mnd  /\  ( F `  x )  e.  ( Base `  ( R `  x )
)  /\  ( G `  x )  e.  (
Base `  ( R `  x ) ) )  ->  ( ( F `
 x ) ( +g  `  ( R `
 x ) ) ( G `  x
) )  e.  (
Base `  ( R `  x ) ) )
2412, 18, 20, 23syl3anc 1184 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
) ( +g  `  ( R `  x )
) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
2524ralrimiva 2733 . . 3  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) ( +g  `  ( R `  x
) ) ( G `
 x ) )  e.  ( Base `  ( R `  x )
) )
261, 2, 3, 4, 7prdsbasmpt 13620 . . 3  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( F `
 x ) ( +g  `  ( R `
 x ) ) ( G `  x
) ) )  e.  B  <->  A. x  e.  I 
( ( F `  x ) ( +g  `  ( R `  x
) ) ( G `
 x ) )  e.  ( Base `  ( R `  x )
) ) )
2725, 26mpbird 224 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) ( +g  `  ( R `  x
) ) ( G `
 x ) ) )  e.  B )
2811, 27eqeltrd 2462 1  |-  ( ph  ->  ( F  .+  G
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650    e. cmpt 4208    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   X_scprds 13597   Mndcmnd 14612
This theorem is referenced by:  prdsmndd  14656  prdsrngd  15646  dsmmacl  26877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-hom 13481  df-cco 13482  df-prds 13599  df-mnd 14618
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