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Theorem pwsdiagrhm 16901
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 16710 . . 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 16653 . . . 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 15777 . . . 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 2443 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
1110rngmgp 16654 . . . . 5  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
12 eqid 2443 . . . . . 6  |-  ( (mulGrp `  R )  ^s  I )  =  ( (mulGrp `  R )  ^s  I )
1310, 6mgpbas 16600 . . . . . 6  |-  B  =  ( Base `  (mulGrp `  R ) )
1412, 13, 7pwsdiagmhm 15500 . . . . 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 2444 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  R
) )  =  (
Base `  (mulGrp `  R
) ) )
17 eqidd 2444 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( Base `  (mulGrp `  Y
) )  =  (
Base `  (mulGrp `  Y
) ) )
18 eqid 2443 . . . . . . 7  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
19 eqid 2443 . . . . . . 7  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
20 eqid 2443 . . . . . . 7  |-  ( Base `  ( (mulGrp `  R
)  ^s  I ) )  =  ( Base `  (
(mulGrp `  R )  ^s  I ) )
21 eqid 2443 . . . . . . 7  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
22 eqid 2443 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)  ^s  I ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  I ) )
232, 10, 12, 18, 19, 20, 21, 22pwsmgp 16713 . . . . . 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 2444 . . . . 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 14632 . . . . 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 15473 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  (
(mulGrp `  R ) MndHom  (mulGrp `  Y ) )  =  ( (mulGrp `  R
) MndHom  ( (mulGrp `  R
)  ^s  I ) ) )
2915, 28eleqtrrd 2520 . . 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 16814 . 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 3880    e. cmpt 4353    X. cxp 4841   ` cfv 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241    ^s cpws 14388   Mndcmnd 15412   Grpcgrp 15413   MndHom cmhm 15465    GrpHom cghm 15747  mulGrpcmgp 16594   Ringcrg 16648   RingHom crh 16807
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 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-fz 11441  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-plusg 14254  df-mulr 14255  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-hom 14265  df-cco 14266  df-0g 14383  df-prds 14389  df-pws 14391  df-mnd 15418  df-mhm 15467  df-grp 15548  df-minusg 15549  df-ghm 15748  df-mgp 16595  df-ur 16607  df-rng 16650  df-rnghom 16809
This theorem is referenced by:  evlsval2  17609
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