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Theorem pwsdiagmhm 14768
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 444 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  R  e.  Mnd )
2 pwsdiagmhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsmnd 14730 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  Y  e.  Mnd )
41, 3jca 519 . 2  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( R  e.  Mnd  /\  Y  e.  Mnd )
)
5 pwsdiagmhm.b . . . . . . 7  |-  B  =  ( Base `  R
)
6 fvex 5742 . . . . . . 7  |-  ( Base `  R )  e.  _V
75, 6eqeltri 2506 . . . . . 6  |-  B  e. 
_V
8 pwsdiagmhm.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
98fdiagfn 7057 . . . . . 6  |-  ( ( B  e.  _V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
107, 9mpan 652 . . . . 5  |-  ( I  e.  W  ->  F : B --> ( B  ^m  I ) )
1110adantl 453 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
122, 5pwsbas 13709 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( B  ^m  I
)  =  ( Base `  Y ) )
13 feq3 5578 . . . . 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 202 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( Base `  Y ) )
16 simplr 732 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  I  e.  W )
17 eqid 2436 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
185, 17mndcl 14695 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
19183expb 1154 . . . . . . 7  |-  ( ( R  e.  Mnd  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  R
) b )  e.  B )
2019adantlr 696 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( +g  `  R ) b )  e.  B )
218fvdiagfn 7058 . . . . . 6  |-  ( ( I  e.  W  /\  ( a ( +g  `  R ) b )  e.  B )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
2216, 20, 21syl2anc 643 . . . . 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 7058 . . . . . . . . 9  |-  ( ( I  e.  W  /\  a  e.  B )  ->  ( F `  a
)  =  ( I  X.  { a } ) )
248fvdiagfn 7058 . . . . . . . . 9  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2523, 24oveqan12d 6100 . . . . . . . 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 804 . . . . . . 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 695 . . . . . 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 2436 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
29 simpll 731 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  Mnd )
302, 5, 28pwsdiagel 13719 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  a  e.  B
)  ->  ( I  X.  { a } )  e.  ( Base `  Y
) )
3130adantrr 698 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
a } )  e.  ( Base `  Y
) )
322, 5, 28pwsdiagel 13719 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  b  e.  B
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
3332adantrl 697 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
b } )  e.  ( Base `  Y
) )
34 eqid 2436 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
352, 28, 29, 16, 31, 33, 17, 34pwsplusgval 13712 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( I  X.  { a } ) ( +g  `  Y
) ( I  X.  { b } ) )  =  ( ( I  X.  { a } )  o F ( +g  `  R
) ( I  X.  { b } ) ) )
36 id 20 . . . . . . . 8  |-  ( I  e.  W  ->  I  e.  W )
37 vex 2959 . . . . . . . . 9  |-  a  e. 
_V
3837a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  a  e.  _V )
39 vex 2959 . . . . . . . . 9  |-  b  e. 
_V
4039a1i 11 . . . . . . . 8  |-  ( I  e.  W  ->  b  e.  _V )
4136, 38, 40ofc12 6329 . . . . . . 7  |-  ( I  e.  W  ->  (
( I  X.  {
a } )  o F ( +g  `  R
) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4241ad2antlr 708 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( I  X.  { a } )  o F ( +g  `  R ) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4327, 35, 423eqtrd 2472 . . . . 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 2471 . . . 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 2798 . . 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 448 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  I  e.  W )
47 eqid 2436 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
485, 47mndidcl 14714 . . . . . 6  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  B )
4948adantr 452 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( 0g `  R
)  e.  B )
508fvdiagfn 7058 . . . . 5  |-  ( ( I  e.  W  /\  ( 0g `  R )  e.  B )  -> 
( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
5146, 49, 50syl2anc 643 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
522, 47pws0g 14731 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  Y
) )
5351, 52eqtrd 2468 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  Y ) )
5415, 45, 533jca 1134 . 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 2436 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
565, 28, 17, 34, 47, 55ismhm 14740 . 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 646 1  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   {csn 3814    e. cmpt 4266    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ^m cmap 7018   Basecbs 13469   +g cplusg 13529    ^s cpws 13670   0gc0g 13723   Mndcmnd 14684   MndHom cmhm 14736
This theorem is referenced by:  pwsdiagghm  15033  pwsdiagrhm  15901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-mnd 14690  df-mhm 14738
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