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Theorem sralmod 17701
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 2441 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 17298 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
52, 4srabase 17692 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  A )
)
62, 4sraaddg 17693 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
72, 4srasca 17695 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
82, 4sravsca 17696 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .s `  A ) )
9 eqid 2441 . . 3  |-  ( Ws  S )  =  ( Ws  S )
109, 3ressbas 14559 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( S  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  S
) ) )
11 eqid 2441 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
129, 11ressplusg 14611 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  ( Ws  S ) ) )
13 eqid 2441 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
149, 13ressmulr 14622 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  ( Ws  S ) ) )
15 eqid 2441 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
169, 15subrg1 17307 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  =  ( 1r `  ( Ws  S ) ) )
179subrgring 17300 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  e.  Ring )
18 subrgrcl 17302 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Ring )
19 ringgrp 17071 . . . 4  |-  ( W  e.  Ring  ->  W  e. 
Grp )
2018, 19syl 16 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Grp )
21 eqidd 2442 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  W )
)
226oveqdr 6301 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
2321, 5, 22grppropd 15937 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  ( W  e.  Grp  <->  A  e.  Grp ) )
2420, 23mpbid 210 . 2  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  Grp )
25183ad2ant1 1016 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  W  e.  Ring )
26 inss2 3701 . . . . 5  |-  ( S  i^i  ( Base `  W
) )  C_  ( Base `  W )
2726sseli 3482 . . . 4  |-  ( x  e.  ( S  i^i  ( Base `  W )
)  ->  x  e.  ( Base `  W )
)
28273ad2ant2 1017 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  x  e.  ( Base `  W )
)
29 simp3 997 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  y  e.  ( Base `  W )
)
303, 13ringcl 17080 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3125, 28, 29, 30syl3anc 1227 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3218adantr 465 . . 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 1001 . . . 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 3484 . . 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 1002 . . 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 1003 . . 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, 13ringdi 17085 . . 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 1229 . 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 465 . . 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 1001 . . . 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 3484 . . 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 1002 . . . 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 3484 . . 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 1003 . . 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, 13ringdir 17086 . . 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 1229 . 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, 13ringass 17083 . . 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 1229 . 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, 15ringlidm 17090 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) x )  =  x )
5018, 49sylan 471 . 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 17386 1  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    i^i cin 3457   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   .rcmulr 14570   Grpcgrp 15922   1rcur 17021   Ringcrg 17066  SubRingcsubrg 17293   LModclmod 17380  subringAlg csra 17682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-subg 16067  df-mgp 17010  df-ur 17022  df-ring 17068  df-subrg 17295  df-lmod 17382  df-sra 17686
This theorem is referenced by:  rlmlmod  17719  sraassa  17842  sranlm  21059
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