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Theorem pwsco1rhm 16829
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 16710 . . . 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 )
87pwsrng 16710 . . . 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 rngmnd 16657 . . . . . 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 15501 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z MndHom  Y ) )
16 rnggrp 16653 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
175, 16syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
18 rnggrp 16653 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
199, 18syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
20 ghmmhmb 15761 . . . . 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 2520 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y ) )
23 eqid 2443 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
24 eqid 2443 . . . . 5  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
25 eqid 2443 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
26 eqid 2443 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
2726rngmgp 16654 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
281, 27syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
2923, 24, 25, 28, 6, 2, 14pwsco1mhm 15501 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  B ) )  |->  ( g  o.  F ) )  e.  ( ( (mulGrp `  R )  ^s  B ) MndHom 
( (mulGrp `  R
)  ^s  A ) ) )
30 eqid 2443 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
313, 30pwsbas 14428 . . . . . . . 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 2493 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  C )
3426, 30mgpbas 16600 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3524, 34pwsbas 14428 . . . . . . 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 2477 . . . . 5  |-  ( ph  ->  C  =  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
3837mpteq1d 4376 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  |->  ( g  o.  F ) ) )
39 eqidd 2444 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
40 eqidd 2444 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
41 eqid 2443 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
42 eqid 2443 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
43 eqid 2443 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
44 eqid 2443 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
453, 26, 24, 41, 42, 25, 43, 44pwsmgp 16713 . . . . . . 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 2443 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2443 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2443 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
51 eqid 2443 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
52 eqid 2443 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
537, 26, 23, 48, 49, 50, 51, 52pwsmgp 16713 . . . . . . 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 14632 . . . . 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 14632 . . . . 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 15473 . . . 4  |-  ( ph  ->  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
)  =  ( ( (mulGrp `  R )  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
6129, 38, 603eltr4d 2524 . . 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 16814 . 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 1369    e. wcel 1756    e. cmpt 4353    o. ccom 4847   -->wf 5417   ` cfv 5421  (class class class)co 6094    ^m cmap 7217   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:  evls1rhmlem  17759
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