MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwsco1rhm Structured version   Unicode version

Theorem pwsco1rhm 17258
Description: Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pwsco1rhm.y  |-  Y  =  ( R  ^s  A )
pwsco1rhm.z  |-  Z  =  ( R  ^s  B )
pwsco1rhm.c  |-  C  =  ( Base `  Z
)
pwsco1rhm.r  |-  ( ph  ->  R  e.  Ring )
pwsco1rhm.a  |-  ( ph  ->  A  e.  V )
pwsco1rhm.b  |-  ( ph  ->  B  e.  W )
pwsco1rhm.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
pwsco1rhm  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z RingHom  Y ) )
Distinct variable groups:    A, g    B, g    ph, g    R, g   
g, Y    C, g    g, F    g, Z
Allowed substitution hints:    V( g)    W( g)

Proof of Theorem pwsco1rhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsco1rhm.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 pwsco1rhm.b . . . 4  |-  ( ph  ->  B  e.  W )
3 pwsco1rhm.z . . . . 5  |-  Z  =  ( R  ^s  B )
43pwsring 17136 . . . 4  |-  ( ( R  e.  Ring  /\  B  e.  W )  ->  Z  e.  Ring )
51, 2, 4syl2anc 661 . . 3  |-  ( ph  ->  Z  e.  Ring )
6 pwsco1rhm.a . . . 4  |-  ( ph  ->  A  e.  V )
7 pwsco1rhm.y . . . . 5  |-  Y  =  ( R  ^s  A )
87pwsring 17136 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
91, 6, 8syl2anc 661 . . 3  |-  ( ph  ->  Y  e.  Ring )
105, 9jca 532 . 2  |-  ( ph  ->  ( Z  e.  Ring  /\  Y  e.  Ring )
)
11 pwsco1rhm.c . . . . 5  |-  C  =  ( Base `  Z
)
12 ringmnd 17079 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
131, 12syl 16 . . . . 5  |-  ( ph  ->  R  e.  Mnd )
14 pwsco1rhm.f . . . . 5  |-  ( ph  ->  F : A --> B )
157, 3, 11, 13, 6, 2, 14pwsco1mhm 15873 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z MndHom  Y ) )
16 ringgrp 17075 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
175, 16syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
18 ringgrp 17075 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
199, 18syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
20 ghmmhmb 16150 . . . . 5  |-  ( ( Z  e.  Grp  /\  Y  e.  Grp )  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2117, 19, 20syl2anc 661 . . . 4  |-  ( ph  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2215, 21eleqtrrd 2558 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y ) )
23 eqid 2467 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
24 eqid 2467 . . . . 5  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
25 eqid 2467 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
26 eqid 2467 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
2726ringmgp 17076 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
281, 27syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
2923, 24, 25, 28, 6, 2, 14pwsco1mhm 15873 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  B ) )  |->  ( g  o.  F ) )  e.  ( ( (mulGrp `  R )  ^s  B ) MndHom 
( (mulGrp `  R
)  ^s  A ) ) )
30 eqid 2467 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
313, 30pwsbas 14759 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  B  e.  W )  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
3213, 2, 31syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
3332, 11syl6eqr 2526 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  C )
3426, 30mgpbas 17019 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3524, 34pwsbas 14759 . . . . . . 7  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  B  e.  W )  ->  (
( Base `  R )  ^m  B )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
3628, 2, 35syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
3733, 36eqtr3d 2510 . . . . 5  |-  ( ph  ->  C  =  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
3837mpteq1d 4534 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  |->  ( g  o.  F ) ) )
39 eqidd 2468 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
40 eqidd 2468 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
41 eqid 2467 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
42 eqid 2467 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
43 eqid 2467 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
44 eqid 2467 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
453, 26, 24, 41, 42, 25, 43, 44pwsmgp 17139 . . . . . . 7  |-  ( ( R  e.  Ring  /\  B  e.  W )  ->  (
( Base `  (mulGrp `  Z
) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  B ) ) ) )
461, 2, 45syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4746simpld 459 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
48 eqid 2467 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2467 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2467 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
51 eqid 2467 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
52 eqid 2467 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
537, 26, 23, 48, 49, 50, 51, 52pwsmgp 17139 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) ) )
541, 6, 53syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5554simpld 459 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
5646simprd 463 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) )
5756proplem3 14963 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Z ) )  /\  y  e.  ( Base `  (mulGrp `  Z )
) ) )  -> 
( x ( +g  `  (mulGrp `  Z )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  B ) ) y ) )
5854simprd 463 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
5958proplem3 14963 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Y ) )  /\  y  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( x ( +g  `  (mulGrp `  Y )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) y ) )
6039, 40, 47, 55, 57, 59mhmpropd 15845 . . . 4  |-  ( ph  ->  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
)  =  ( ( (mulGrp `  R )  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
6129, 38, 603eltr4d 2570 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( (mulGrp `  Z ) MndHom  (mulGrp `  Y ) ) )
6222, 61jca 532 . 2  |-  ( ph  ->  ( ( g  e.  C  |->  ( g  o.  F ) )  e.  ( Z  GrpHom  Y )  /\  ( g  e.  C  |->  ( g  o.  F ) )  e.  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
) ) )
6341, 48isrhm 17242 . 2  |-  ( ( g  e.  C  |->  ( g  o.  F ) )  e.  ( Z RingHom  Y )  <->  ( ( Z  e.  Ring  /\  Y  e.  Ring )  /\  (
( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y )  /\  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( (mulGrp `  Z ) MndHom  (mulGrp `  Y ) ) ) ) )
6410, 62, 63sylanbrc 664 1  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z RingHom  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Basecbs 14507   +g cplusg 14572    ^s cpws 14719   Mndcmnd 15793   MndHom cmhm 15837   Grpcgrp 15925    GrpHom cghm 16136  mulGrpcmgp 17013   Ringcrg 17070   RingHom crh 17233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-rnghom 17236
This theorem is referenced by:  evls1rhmlem  18228
  Copyright terms: Public domain W3C validator