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Theorem sralmod 16260
Description: The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
sralmod.a  |-  A  =  ( ( subringAlg  `  W ) `
 S )
Assertion
Ref Expression
sralmod  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )

Proof of Theorem sralmod
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sralmod.a . . . 4  |-  A  =  ( ( subringAlg  `  W ) `
 S )
21a1i 11 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
3 eqid 2438 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 15871 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
52, 4srabase 16252 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  A )
)
62, 4sraaddg 16253 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
72, 4srasca 16255 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
82, 4sravsca 16256 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .s `  A ) )
9 eqid 2438 . . 3  |-  ( Ws  S )  =  ( Ws  S )
109, 3ressbas 13521 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( S  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  S
) ) )
11 eqid 2438 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
129, 11ressplusg 13573 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  ( Ws  S ) ) )
13 eqid 2438 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
149, 13ressmulr 13584 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  ( Ws  S ) ) )
15 eqid 2438 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
169, 15subrg1 15880 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  =  ( 1r `  ( Ws  S ) ) )
179subrgrng 15873 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  e.  Ring )
18 subrgrcl 15875 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Ring )
19 rnggrp 15671 . . . 4  |-  ( W  e.  Ring  ->  W  e. 
Grp )
2018, 19syl 16 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Grp )
21 eqidd 2439 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  W )
)
226proplem3 13918 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
2321, 5, 22grppropd 14825 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  ( W  e.  Grp  <->  A  e.  Grp ) )
2420, 23mpbid 203 . 2  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  Grp )
25183ad2ant1 979 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  W  e.  Ring )
26 inss2 3564 . . . . 5  |-  ( S  i^i  ( Base `  W
) )  C_  ( Base `  W )
2726sseli 3346 . . . 4  |-  ( x  e.  ( S  i^i  ( Base `  W )
)  ->  x  e.  ( Base `  W )
)
28273ad2ant2 980 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  x  e.  ( Base `  W )
)
29 simp3 960 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  y  e.  ( Base `  W )
)
303, 13rngcl 15679 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3125, 28, 29, 30syl3anc 1185 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3218adantr 453 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  W  e.  Ring )
33 simpr1 964 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  x  e.  ( S  i^i  ( Base `  W ) ) )
3426, 33sseldi 3348 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  x  e.  ( Base `  W )
)
35 simpr2 965 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  y  e.  ( Base `  W )
)
36 simpr3 966 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  z  e.  ( Base `  W )
)
373, 11, 13rngdi 15684 . . 3  |-  ( ( W  e.  Ring  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
)  /\  z  e.  ( Base `  W )
) )  ->  (
x ( .r `  W ) ( y ( +g  `  W
) z ) )  =  ( ( x ( .r `  W
) y ) ( +g  `  W ) ( x ( .r
`  W ) z ) ) )
3832, 34, 35, 36, 37syl13anc 1187 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )  /\  z  e.  ( Base `  W ) ) )  ->  ( x
( .r `  W
) ( y ( +g  `  W ) z ) )  =  ( ( x ( .r `  W ) y ) ( +g  `  W ) ( x ( .r `  W
) z ) ) )
3918adantr 453 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  W  e.  Ring )
40 simpr1 964 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  x  e.  ( S  i^i  ( Base `  W ) ) )
4126, 40sseldi 3348 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  x  e.  (
Base `  W )
)
42 simpr2 965 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  y  e.  ( S  i^i  ( Base `  W ) ) )
4326, 42sseldi 3348 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  y  e.  (
Base `  W )
)
44 simpr3 966 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  z  e.  (
Base `  W )
)
453, 11, 13rngdir 15685 . . 3  |-  ( ( W  e.  Ring  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
)  /\  z  e.  ( Base `  W )
) )  ->  (
( x ( +g  `  W ) y ) ( .r `  W
) z )  =  ( ( x ( .r `  W ) z ) ( +g  `  W ) ( y ( .r `  W
) z ) ) )
4639, 41, 43, 44, 45syl13anc 1187 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  ( ( x ( +g  `  W
) y ) ( .r `  W ) z )  =  ( ( x ( .r
`  W ) z ) ( +g  `  W
) ( y ( .r `  W ) z ) ) )
473, 13rngass 15682 . . 3  |-  ( ( W  e.  Ring  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
)  /\  z  e.  ( Base `  W )
) )  ->  (
( x ( .r
`  W ) y ) ( .r `  W ) z )  =  ( x ( .r `  W ) ( y ( .r
`  W ) z ) ) )
4839, 41, 43, 44, 47syl13anc 1187 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  ( S  i^i  ( Base `  W ) )  /\  z  e.  ( Base `  W ) ) )  ->  ( ( x ( .r `  W
) y ) ( .r `  W ) z )  =  ( x ( .r `  W ) ( y ( .r `  W
) z ) ) )
493, 13, 15rnglidm 15689 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) x )  =  x )
5018, 49sylan 459 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( Base `  W )
)  ->  ( ( 1r `  W ) ( .r `  W ) x )  =  x )
515, 6, 7, 8, 10, 12, 14, 16, 17, 24, 31, 38, 46, 48, 50islmodd 15958 1  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321   ` cfv 5456  (class class class)co 6083   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   .rcmulr 13532   Grpcgrp 14687   Ringcrg 15662   1rcur 15664  SubRingcsubrg 15866   LModclmod 15952   subringAlg csra 16242
This theorem is referenced by:  rlmlmod  16278  sraassa  16386  sranlm  18722
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-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-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  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-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-mnd 14692  df-grp 14814  df-subg 14943  df-mgp 15651  df-rng 15665  df-ur 15667  df-subrg 15868  df-lmod 15954  df-sra 16246
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