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Theorem pwsdiagmhm 14445
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 14407 . . 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 5539 . . . . . . 7  |-  ( Base `  R )  e.  _V
75, 6eqeltri 2353 . . . . . 6  |-  B  e. 
_V
8 pwsdiagmhm.f . . . . . . 7  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
98fdiagfn 6811 . . . . . 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 13386 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( B  ^m  I
)  =  ( Base `  Y ) )
13 feq3 5377 . . . . 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 2283 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
185, 17mndcl 14372 . . . . . . . 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 6812 . . . . . 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 6812 . . . . . . . . 9  |-  ( ( I  e.  W  /\  a  e.  B )  ->  ( F `  a
)  =  ( I  X.  { a } ) )
248fvdiagfn 6812 . . . . . . . . 9  |-  ( ( I  e.  W  /\  b  e.  B )  ->  ( F `  b
)  =  ( I  X.  { b } ) )
2523, 24oveqan12d 5877 . . . . . . . 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 2283 . . . . . . 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 13396 . . . . . . . 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 13396 . . . . . . . 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 2283 . . . . . . 7  |-  ( +g  `  Y )  =  ( +g  `  Y )
352, 28, 29, 16, 31, 33, 17, 34pwsplusgval 13389 . . . . . 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 2791 . . . . . . . . 9  |-  a  e. 
_V
3837a1i 10 . . . . . . . 8  |-  ( I  e.  W  ->  a  e.  _V )
39 vex 2791 . . . . . . . . 9  |-  b  e. 
_V
4039a1i 10 . . . . . . . 8  |-  ( I  e.  W  ->  b  e.  _V )
4136, 38, 40ofc12 6102 . . . . . . 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 2319 . . . . 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 2318 . . . 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 2635 . . 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 2283 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
485, 47mndidcl 14391 . . . . . 6  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  B )
4948adantr 451 . . . . 5  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( 0g `  R
)  e.  B )
508fvdiagfn 6812 . . . . 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 14408 . . . 4  |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  ( I  X.  {
( 0g `  R
) } )  =  ( 0g `  Y
) )
5351, 52eqtrd 2315 . . 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 2283 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
565, 28, 17, 34, 47, 55ismhm 14417 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Basecbs 13148   +g cplusg 13208    ^s cpws 13347   0gc0g 13400   Mndcmnd 14361   MndHom cmhm 14413
This theorem is referenced by:  pwsdiagghm  14710  pwsdiagrhm  15578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-mnd 14367  df-mhm 14415
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