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Theorem pwsco1rhm 15835
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 )
43pwsrng 15723 . . . 4  |-  ( ( R  e.  Ring  /\  B  e.  W )  ->  Z  e.  Ring )
51, 2, 4syl2anc 644 . . 3  |-  ( ph  ->  Z  e.  Ring )
6 pwsco1rhm.a . . . 4  |-  ( ph  ->  A  e.  V )
7 pwsco1rhm.y . . . . 5  |-  Y  =  ( R  ^s  A )
87pwsrng 15723 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
91, 6, 8syl2anc 644 . . 3  |-  ( ph  ->  Y  e.  Ring )
105, 9jca 520 . 2  |-  ( ph  ->  ( Z  e.  Ring  /\  Y  e.  Ring )
)
11 pwsco1rhm.c . . . . 5  |-  C  =  ( Base `  Z
)
12 rngmnd 15675 . . . . . 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 14771 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z MndHom  Y ) )
16 rnggrp 15671 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
175, 16syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
18 rnggrp 15671 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
199, 18syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
20 ghmmhmb 15019 . . . . 5  |-  ( ( Z  e.  Grp  /\  Y  e.  Grp )  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2117, 19, 20syl2anc 644 . . . 4  |-  ( ph  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2215, 21eleqtrrd 2515 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y ) )
23 eqid 2438 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
24 eqid 2438 . . . . 5  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
25 eqid 2438 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
26 eqid 2438 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
2726rngmgp 15672 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
281, 27syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
2923, 24, 25, 28, 6, 2, 14pwsco1mhm 14771 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  B ) )  |->  ( g  o.  F ) )  e.  ( ( (mulGrp `  R )  ^s  B ) MndHom 
( (mulGrp `  R
)  ^s  A ) ) )
30 eqid 2438 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
313, 30pwsbas 13711 . . . . . . . 8  |-  ( ( R  e.  Mnd  /\  B  e.  W )  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
3213, 2, 31syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
3332, 11syl6eqr 2488 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  C )
3426, 30mgpbas 15656 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3524, 34pwsbas 13711 . . . . . . 7  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  B  e.  W )  ->  (
( Base `  R )  ^m  B )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
3628, 2, 35syl2anc 644 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
3733, 36eqtr3d 2472 . . . . 5  |-  ( ph  ->  C  =  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
3837mpteq1d 4292 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  |->  ( g  o.  F ) ) )
39 eqidd 2439 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
40 eqidd 2439 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
41 eqid 2438 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
42 eqid 2438 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
43 eqid 2438 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
44 eqid 2438 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
453, 26, 24, 41, 42, 25, 43, 44pwsmgp 15726 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4746simpld 447 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
48 eqid 2438 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2438 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2438 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
51 eqid 2438 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
52 eqid 2438 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
537, 26, 23, 48, 49, 50, 51, 52pwsmgp 15726 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5554simpld 447 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
5646simprd 451 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) )
5756proplem3 13918 . . . . 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 451 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
5958proplem3 13918 . . . . 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 14746 . . . 4  |-  ( ph  ->  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
)  =  ( ( (mulGrp `  R )  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
6129, 38, 603eltr4d 2519 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( (mulGrp `  Z ) MndHom  (mulGrp `  Y ) ) )
6222, 61jca 520 . 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 15826 . 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 647 1  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z RingHom  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   Basecbs 13471   +g cplusg 13531    ^s cpws 13672   Mndcmnd 14686   Grpcgrp 14687   MndHom cmhm 14738    GrpHom cghm 15005  mulGrpcmgp 15650   Ringcrg 15662   RingHom crh 15819
This theorem is referenced by:  evl1rhm  19951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-pws 13675  df-0g 13729  df-mnd 14692  df-mhm 14740  df-grp 14814  df-minusg 14815  df-ghm 15006  df-mgp 15651  df-rng 15665  df-ur 15667  df-rnghom 15821
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