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Theorem pwsdiagmhm 14494
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 443 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  R  e.  Mnd )
2 pwsdiagmhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsmnd 14456 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  Y  e.  Mnd )
41, 3jca 518 . 2  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( R  e.  Mnd  /\  Y  e.  Mnd )
)
5 pwsdiagmhm.b . . . . . . 7  |-  B  =  ( Base `  R
)
6 fvex 5577 . . . . . . 7  |-  ( Base `  R )  e.  _V
75, 6eqeltri 2386 . . . . . 6  |-  B  e. 
_V
8 pwsdiagmhm.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
98fdiagfn 6854 . . . . . 6  |-  ( ( B  e.  _V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
107, 9mpan 651 . . . . 5  |-  ( I  e.  W  ->  F : B --> ( B  ^m  I ) )
1110adantl 452 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
122, 5pwsbas 13435 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( B  ^m  I
)  =  ( Base `  Y ) )
13 feq3 5414 . . . . 5  |-  ( ( B  ^m  I )  =  ( Base `  Y
)  ->  ( F : B --> ( B  ^m  I )  <->  F : B
--> ( Base `  Y
) ) )
1412, 13syl 15 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F : B --> ( B  ^m  I )  <-> 
F : B --> ( Base `  Y ) ) )
1511, 14mpbid 201 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F : B --> ( Base `  Y ) )
16 simplr 731 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  I  e.  W )
17 eqid 2316 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
185, 17mndcl 14421 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
19183expb 1152 . . . . . . 7  |-  ( ( R  e.  Mnd  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  R
) b )  e.  B )
2019adantlr 695 . . . . . 6  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( +g  `  R ) b )  e.  B )
218fvdiagfn 6855 . . . . . 6  |-  ( ( I  e.  W  /\  ( a ( +g  `  R ) b )  e.  B )  -> 
( F `  (
a ( +g  `  R
) b ) )  =  ( I  X.  { ( a ( +g  `  R ) b ) } ) )
2216, 20, 21syl2anc 642 . . . . 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 6855 . . . . . . . . 9  |-  ( ( I  e.  W  /\  a  e.  B )  ->  ( F `  a
)  =  ( I  X.  { a } ) )
248fvdiagfn 6855 . . . . . . . . 9  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2523, 24oveqan12d 5919 . . . . . . . 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 803 . . . . . . 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 694 . . . . . 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 2316 . . . . . . 7  |-  ( Base `  Y )  =  (
Base `  Y )
29 simpll 730 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  ->  R  e.  Mnd )
302, 5, 28pwsdiagel 13445 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  a  e.  B
)  ->  ( I  X.  { a } )  e.  ( Base `  Y
) )
3130adantrr 697 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
a } )  e.  ( Base `  Y
) )
322, 5, 28pwsdiagel 13445 . . . . . . . 8  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  b  e.  B
)  ->  ( I  X.  { b } )  e.  ( Base `  Y
) )
3332adantrl 696 . . . . . . 7  |-  ( ( ( R  e.  Mnd  /\  I  e.  W )  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( I  X.  {
b } )  e.  ( Base `  Y
) )
34 eqid 2316 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
352, 28, 29, 16, 31, 33, 17, 34pwsplusgval 13438 . . . . . 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 19 . . . . . . . 8  |-  ( I  e.  W  ->  I  e.  W )
37 vex 2825 . . . . . . . . 9  |-  a  e. 
_V
3837a1i 10 . . . . . . . 8  |-  ( I  e.  W  ->  a  e.  _V )
39 vex 2825 . . . . . . . . 9  |-  b  e. 
_V
4039a1i 10 . . . . . . . 8  |-  ( I  e.  W  ->  b  e.  _V )
4136, 38, 40ofc12 6144 . . . . . . 7  |-  ( I  e.  W  ->  (
( I  X.  {
a } )  o F ( +g  `  R
) ( I  X.  { b } ) )  =  ( I  X.  { ( a ( +g  `  R
) b ) } ) )
4241ad2antlr 707 . . . . . 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 2352 . . . . 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 2351 . . . 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 2669 . . 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 447 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  I  e.  W )
47 eqid 2316 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
485, 47mndidcl 14440 . . . . . 6  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  B )
4948adantr 451 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( 0g `  R
)  e.  B )
508fvdiagfn 6855 . . . . 5  |-  ( ( I  e.  W  /\  ( 0g `  R )  e.  B )  -> 
( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
5146, 49, 50syl2anc 642 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( I  X.  { ( 0g `  R ) } ) )
522, 47pws0g 14457 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  Y
) )
5351, 52eqtrd 2348 . . 3  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  Y ) )
5415, 45, 533jca 1132 . 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 2316 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
565, 28, 17, 34, 47, 55ismhm 14466 . 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 645 1  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822   {csn 3674    e. cmpt 4114    X. cxp 4724   -->wf 5288   ` cfv 5292  (class class class)co 5900    o Fcof 6118    ^m cmap 6815   Basecbs 13195   +g cplusg 13255    ^s cpws 13396   0gc0g 13449   Mndcmnd 14410   MndHom cmhm 14462
This theorem is referenced by:  pwsdiagghm  14759  pwsdiagrhm  15627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-hom 13279  df-cco 13280  df-prds 13397  df-pws 13399  df-0g 13453  df-mnd 14416  df-mhm 14464
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