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Theorem pwsdiagmhm 15492
Description: Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
Hypotheses
Ref Expression
pwsdiagmhm.y  |-  Y  =  ( R  ^s  I )
pwsdiagmhm.b  |-  B  =  ( Base `  R
)
pwsdiagmhm.f  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
Assertion
Ref Expression
pwsdiagmhm  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y ) )
Distinct variable groups:    x, Y    x, R    x, I    x, B    x, W
Allowed substitution hint:    F( x)

Proof of Theorem pwsdiagmhm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 454 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  R  e.  Mnd )
2 pwsdiagmhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsmnd 15452 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  Y  e.  Mnd )
41, 3jca 529 . 2  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( R  e.  Mnd  /\  Y  e.  Mnd )
)
5 pwsdiagmhm.b . . . . . . 7  |-  B  =  ( Base `  R
)
6 fvex 5698 . . . . . . 7  |-  ( Base `  R )  e.  _V
75, 6eqeltri 2511 . . . . . 6  |-  B  e. 
_V
8 pwsdiagmhm.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
98fdiagfn 7252 . . . . . 6  |-  ( ( B  e.  _V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
107, 9mpan 665 . . . . 5  |-  ( I  e.  W  ->  F : B --> ( B  ^m  I ) )
1110adantl 463 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
122, 5pwsbas 14421 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( B  ^m  I
)  =  ( Base `  Y ) )
13 feq3 5541 . . . . 5  |-  ( ( B  ^m  I )  =  ( Base `  Y
)  ->  ( F : B --> ( B  ^m  I )  <->  F : B
--> ( Base `  Y
) ) )
1412, 13syl 16 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F : B --> ( B  ^m  I )  <-> 
F : B --> ( Base `  Y ) ) )
1511, 14mpbid 210 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( Base `  Y ) )
16 simplr 749 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  I  e.  W )
17 eqid 2441 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
185, 17mndcl 15416 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
19183expb 1183 . . . . . . 7  |-  ( ( R  e.  Mnd  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  R
) b )  e.  B )
2019adantlr 709 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( +g  `  R ) b )  e.  B )
218fvdiagfn 7253 . . . . . 6  |-  ( ( I  e.  W  /\  ( a ( +g  `  R ) b )  e.  B )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
2216, 20, 21syl2anc 656 . . . . 5  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
238fvdiagfn 7253 . . . . . . . . 9  |-  ( ( I  e.  W  /\  a  e.  B )  ->  ( F `  a
)  =  ( I  X.  { a } ) )
248fvdiagfn 7253 . . . . . . . . 9  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2523, 24oveqan12d 6109 . . . . . . . 8  |-  ( ( ( I  e.  W  /\  a  e.  B
)  /\  ( I  e.  W  /\  b  e.  B ) )  -> 
( ( F `  a ) ( +g  `  Y ) ( F `
 b ) )  =  ( ( I  X.  { a } ) ( +g  `  Y
) ( I  X.  { b } ) ) )
2625anandis 821 . . . . . . 7  |-  ( ( I  e.  W  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
) ( +g  `  Y
) ( F `  b ) )  =  ( ( I  X.  { a } ) ( +g  `  Y
) ( I  X.  { b } ) ) )
2726adantll 708 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( F `  a ) ( +g  `  Y ) ( F `
 b ) )  =  ( ( I  X.  { a } ) ( +g  `  Y
) ( I  X.  { b } ) ) )
28 eqid 2441 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
29 simpll 748 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  Mnd )
302, 5, 28pwsdiagel 14431 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  a  e.  B
)  ->  ( I  X.  { a } )  e.  ( Base `  Y
) )
3130adantrr 711 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
a } )  e.  ( Base `  Y
) )
322, 5, 28pwsdiagel 14431 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  b  e.  B
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
3332adantrl 710 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
b } )  e.  ( Base `  Y
) )
34 eqid 2441 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
352, 28, 29, 16, 31, 33, 17, 34pwsplusgval 14424 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( I  X.  { a } ) ( +g  `  Y
) ( I  X.  { b } ) )  =  ( ( I  X.  { a } )  oF ( +g  `  R
) ( I  X.  { b } ) ) )
36 id 22 . . . . . . . 8  |-  ( I  e.  W  ->  I  e.  W )
37 vex 2973 . . . . . . . . 9  |-  a  e. 
_V
3837a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  a  e.  _V )
39 vex 2973 . . . . . . . . 9  |-  b  e. 
_V
4039a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  b  e.  _V )
4136, 38, 40ofc12 6344 . . . . . . 7  |-  ( I  e.  W  ->  (
( I  X.  {
a } )  oF ( +g  `  R
) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4241ad2antlr 721 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( I  X.  { a } )  oF ( +g  `  R ) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4327, 35, 423eqtrd 2477 . . . . 5  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( F `  a ) ( +g  `  Y ) ( F `
 b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
4422, 43eqtr4d 2476 . . . 4  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( ( F `
 a ) ( +g  `  Y ) ( F `  b
) ) )
4544ralrimivva 2806 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  A. a  e.  B  A. b  e.  B  ( F `  ( a ( +g  `  R
) b ) )  =  ( ( F `
 a ) ( +g  `  Y ) ( F `  b
) ) )
46 simpr 458 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  I  e.  W )
47 eqid 2441 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
485, 47mndidcl 15435 . . . . . 6  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  B )
4948adantr 462 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( 0g `  R
)  e.  B )
508fvdiagfn 7253 . . . . 5  |-  ( ( I  e.  W  /\  ( 0g `  R )  e.  B )  -> 
( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
5146, 49, 50syl2anc 656 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
522, 47pws0g 15453 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  Y
) )
5351, 52eqtrd 2473 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  Y ) )
5415, 45, 533jca 1163 . 2  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F : B --> ( Base `  Y )  /\  A. a  e.  B  A. b  e.  B  ( F `  ( a ( +g  `  R
) b ) )  =  ( ( F `
 a ) ( +g  `  Y ) ( F `  b
) )  /\  ( F `  ( 0g `  R ) )  =  ( 0g `  Y
) ) )
55 eqid 2441 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
565, 28, 17, 34, 47, 55ismhm 15462 . 2  |-  ( F  e.  ( R MndHom  Y
)  <->  ( ( R  e.  Mnd  /\  Y  e.  Mnd )  /\  ( F : B --> ( Base `  Y )  /\  A. a  e.  B  A. b  e.  B  ( F `  ( a
( +g  `  R ) b ) )  =  ( ( F `  a ) ( +g  `  Y ) ( F `
 b ) )  /\  ( F `  ( 0g `  R ) )  =  ( 0g
`  Y ) ) ) )
574, 54, 56sylanbrc 659 1  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   {csn 3874    e. cmpt 4347    X. cxp 4834   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317    ^m cmap 7210   Basecbs 14170   +g cplusg 14234   0gc0g 14374    ^s cpws 14381   Mndcmnd 15405   MndHom cmhm 15458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-0g 14376  df-prds 14382  df-pws 14384  df-mnd 15411  df-mhm 15460
This theorem is referenced by:  pwsdiagghm  15767  pwsdiagrhm  16878
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