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Theorem pwsdiagrhm 16874
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 457 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  R  e.  Ring )
2 pwsdiagrhm.y . . . 4  |-  Y  =  ( R  ^s  I )
32pwsrng 16693 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  Y  e.  Ring )
41, 3jca 532 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( R  e.  Ring  /\  Y  e.  Ring ) )
5 rnggrp 16636 . . . 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 15763 . . . 4  |-  ( ( R  e.  Grp  /\  I  e.  W )  ->  F  e.  ( R 
GrpHom  Y ) )
95, 8sylan 471 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R  GrpHom  Y ) )
10 eqid 2437 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
1110rngmgp 16637 . . . . 5  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
12 eqid 2437 . . . . . 6  |-  ( (mulGrp `  R )  ^s  I )  =  ( (mulGrp `  R )  ^s  I )
1310, 6mgpbas 16583 . . . . . 6  |-  B  =  ( Base `  (mulGrp `  R ) )
1412, 13, 7pwsdiagmhm 15488 . . . . 5  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
1511, 14sylan 471 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
16 eqidd 2438 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  R
) )  =  (
Base `  (mulGrp `  R
) ) )
17 eqidd 2438 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  Y
) )  =  (
Base `  (mulGrp `  Y
) ) )
18 eqid 2437 . . . . . . 7  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
19 eqid 2437 . . . . . . 7  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
20 eqid 2437 . . . . . . 7  |-  ( Base `  ( (mulGrp `  R
)  ^s  I ) )  =  ( Base `  (
(mulGrp `  R )  ^s  I ) )
21 eqid 2437 . . . . . . 7  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
22 eqid 2437 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)  ^s  I ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  I ) )
232, 10, 12, 18, 19, 20, 21, 22pwsmgp 16696 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  I ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  I ) ) ) )
2423simpld 459 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  I ) ) )
25 eqidd 2438 . . . . 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 463 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( +g  `  (mulGrp `  Y
) )  =  ( +g  `  ( (mulGrp `  R )  ^s  I ) ) )
2726proplem3 14621 . . . . 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 15462 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
(mulGrp `  R ) MndHom  (mulGrp `  Y ) )  =  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
2915, 28eleqtrrd 2514 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  Y )
) )
309, 29jca 532 . 2  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( F  e.  ( R  GrpHom  Y )  /\  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  Y )
) ) )
3110, 18isrhm 16797 . 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 664 1  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R RingHom  Y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3870    e. cmpt 4343    X. cxp 4830   ` cfv 5411  (class class class)co 6086   Basecbs 14166   +g cplusg 14230    ^s cpws 14377   Mndcmnd 15401   Grpcgrp 15402   MndHom cmhm 15454    GrpHom cghm 15733  mulGrpcmgp 16577   Ringcrg 16631   RingHom crh 16790
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-hom 14254  df-cco 14255  df-0g 14372  df-prds 14378  df-pws 14380  df-mnd 15407  df-mhm 15456  df-grp 15534  df-minusg 15535  df-ghm 15734  df-mgp 16578  df-ur 16590  df-rng 16633  df-rnghom 16792
This theorem is referenced by:  evlsval2  17578
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