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Theorem pwsdiagrhm 15578
Description: Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
pwsdiagrhm.y  |-  Y  =  ( R  ^s  I )
pwsdiagrhm.b  |-  B  =  ( Base `  R
)
pwsdiagrhm.f  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
Assertion
Ref Expression
pwsdiagrhm  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R RingHom  Y )
)
Distinct variable groups:    x, B    x, I    x, R    x, W    x, Y
Allowed substitution hint:    F( x)

Proof of Theorem pwsdiagrhm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  R  e.  Ring )
2 pwsdiagrhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsrng 15398 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  Y  e.  Ring )
41, 3jca 518 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( R  e.  Ring  /\  Y  e.  Ring ) )
5 rnggrp 15346 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
6 pwsdiagrhm.b . . . . 5  |-  B  =  ( Base `  R
)
7 pwsdiagrhm.f . . . . 5  |-  F  =  ( x  e.  B  |->  ( I  X.  {
x } ) )
82, 6, 7pwsdiagghm 14710 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  W )  ->  F  e.  ( R 
GrpHom  Y ) )
95, 8sylan 457 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R  GrpHom  Y ) )
10 eqid 2283 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
1110rngmgp 15347 . . . . 5  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
12 eqid 2283 . . . . . 6  |-  ( (mulGrp `  R )  ^s  I )  =  ( (mulGrp `  R )  ^s  I )
1310, 6mgpbas 15331 . . . . . 6  |-  B  =  ( Base `  (mulGrp `  R ) )
1412, 13, 7pwsdiagmhm 14445 . . . . 5  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
1511, 14sylan 457 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
16 eqidd 2284 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  R
) )  =  (
Base `  (mulGrp `  R
) ) )
17 eqidd 2284 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  Y
) )  =  (
Base `  (mulGrp `  Y
) ) )
18 eqid 2283 . . . . . . 7  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
19 eqid 2283 . . . . . . 7  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
20 eqid 2283 . . . . . . 7  |-  ( Base `  ( (mulGrp `  R
)  ^s  I ) )  =  ( Base `  (
(mulGrp `  R )  ^s  I ) )
21 eqid 2283 . . . . . . 7  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
22 eqid 2283 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)  ^s  I ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  I ) )
232, 10, 12, 18, 19, 20, 21, 22pwsmgp 15401 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  I ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  I ) ) ) )
2423simpld 445 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  I ) ) )
25 eqidd 2284 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  ( y  e.  ( Base `  (mulGrp `  R ) )  /\  z  e.  ( Base `  (mulGrp `  R )
) ) )  -> 
( y ( +g  `  (mulGrp `  R )
) z )  =  ( y ( +g  `  (mulGrp `  R )
) z ) )
2623simprd 449 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( +g  `  (mulGrp `  Y
) )  =  ( +g  `  ( (mulGrp `  R )  ^s  I ) ) )
2726proplem3 13593 . . . . 5  |-  ( ( ( R  e.  Ring  /\  I  e.  W )  /\  ( y  e.  ( Base `  (mulGrp `  Y ) )  /\  z  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( y ( +g  `  (mulGrp `  Y )
) z )  =  ( y ( +g  `  ( (mulGrp `  R
)  ^s  I ) ) z ) )
2816, 17, 16, 24, 25, 27mhmpropd 14421 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
(mulGrp `  R ) MndHom  (mulGrp `  Y ) )  =  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
2915, 28eleqtrrd 2360 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  Y )
) )
309, 29jca 518 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( F  e.  ( R  GrpHom  Y )  /\  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  Y )
) ) )
3110, 18isrhm 15501 . 2  |-  ( F  e.  ( R RingHom  Y
)  <->  ( ( R  e.  Ring  /\  Y  e. 
Ring )  /\  ( F  e.  ( R  GrpHom  Y )  /\  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  Y )
) ) ) )
324, 30, 31sylanbrc 645 1  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R RingHom  Y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208    ^s cpws 13347   Mndcmnd 14361   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680  mulGrpcmgp 15325   Ringcrg 15337   RingHom crh 15494
This theorem is referenced by:  evlsval2  19404
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-sets 13154  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  df-grp 14489  df-minusg 14490  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-rnghom 15496
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