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Theorem pwsdiagmhm 15502
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 457 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  R  e.  Mnd )
2 pwsdiagmhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsmnd 15461 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  Y  e.  Mnd )
41, 3jca 532 . 2  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( R  e.  Mnd  /\  Y  e.  Mnd )
)
5 pwsdiagmhm.b . . . . . . 7  |-  B  =  ( Base `  R
)
6 fvex 5706 . . . . . . 7  |-  ( Base `  R )  e.  _V
75, 6eqeltri 2513 . . . . . 6  |-  B  e. 
_V
8 pwsdiagmhm.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
98fdiagfn 7261 . . . . . 6  |-  ( ( B  e.  _V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
107, 9mpan 670 . . . . 5  |-  ( I  e.  W  ->  F : B --> ( B  ^m  I ) )
1110adantl 466 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
122, 5pwsbas 14430 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( B  ^m  I
)  =  ( Base `  Y ) )
13 feq3 5549 . . . . 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 754 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  I  e.  W )
17 eqid 2443 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
185, 17mndcl 15425 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
19183expb 1188 . . . . . . 7  |-  ( ( R  e.  Mnd  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  R
) b )  e.  B )
2019adantlr 714 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( +g  `  R ) b )  e.  B )
218fvdiagfn 7262 . . . . . 6  |-  ( ( I  e.  W  /\  ( a ( +g  `  R ) b )  e.  B )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
2216, 20, 21syl2anc 661 . . . . 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 7262 . . . . . . . . 9  |-  ( ( I  e.  W  /\  a  e.  B )  ->  ( F `  a
)  =  ( I  X.  { a } ) )
248fvdiagfn 7262 . . . . . . . . 9  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2523, 24oveqan12d 6115 . . . . . . . 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 826 . . . . . . 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 713 . . . . . 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 2443 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
29 simpll 753 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  Mnd )
302, 5, 28pwsdiagel 14440 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  a  e.  B
)  ->  ( I  X.  { a } )  e.  ( Base `  Y
) )
3130adantrr 716 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
a } )  e.  ( Base `  Y
) )
322, 5, 28pwsdiagel 14440 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  b  e.  B
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
3332adantrl 715 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
b } )  e.  ( Base `  Y
) )
34 eqid 2443 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
352, 28, 29, 16, 31, 33, 17, 34pwsplusgval 14433 . . . . . 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 2980 . . . . . . . . 9  |-  a  e. 
_V
3837a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  a  e.  _V )
39 vex 2980 . . . . . . . . 9  |-  b  e. 
_V
4039a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  b  e.  _V )
4136, 38, 40ofc12 6350 . . . . . . 7  |-  ( I  e.  W  ->  (
( I  X.  {
a } )  oF ( +g  `  R
) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4241ad2antlr 726 . . . . . 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 2479 . . . . 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 2478 . . . 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 2813 . . 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 461 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  I  e.  W )
47 eqid 2443 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
485, 47mndidcl 15444 . . . . . 6  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  B )
4948adantr 465 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( 0g `  R
)  e.  B )
508fvdiagfn 7262 . . . . 5  |-  ( ( I  e.  W  /\  ( 0g `  R )  e.  B )  -> 
( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
5146, 49, 50syl2anc 661 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
522, 47pws0g 15462 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  Y
) )
5351, 52eqtrd 2475 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  Y ) )
5415, 45, 533jca 1168 . 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 2443 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
565, 28, 17, 34, 47, 55ismhm 15471 . 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 664 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977   {csn 3882    e. cmpt 4355    X. cxp 4843   -->wf 5419   ` cfv 5423  (class class class)co 6096    oFcof 6323    ^m cmap 7219   Basecbs 14179   +g cplusg 14243   0gc0g 14383    ^s cpws 14390   Mndcmnd 15414   MndHom cmhm 15467
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-0g 14385  df-prds 14391  df-pws 14393  df-mnd 15420  df-mhm 15469
This theorem is referenced by:  pwsdiagghm  15779  pwsdiagrhm  16903
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