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Theorem sralmod 17386
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 2452 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 16984 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
52, 4srabase 17377 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  A )
)
62, 4sraaddg 17378 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
72, 4srasca 17380 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
82, 4sravsca 17381 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .s `  A ) )
9 eqid 2452 . . 3  |-  ( Ws  S )  =  ( Ws  S )
109, 3ressbas 14342 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( S  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  S
) ) )
11 eqid 2452 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
129, 11ressplusg 14394 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  ( Ws  S ) ) )
13 eqid 2452 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
149, 13ressmulr 14405 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  ( Ws  S ) ) )
15 eqid 2452 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
169, 15subrg1 16993 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  =  ( 1r `  ( Ws  S ) ) )
179subrgrng 16986 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  e.  Ring )
18 subrgrcl 16988 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Ring )
19 rnggrp 16768 . . . 4  |-  ( W  e.  Ring  ->  W  e. 
Grp )
2018, 19syl 16 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Grp )
21 eqidd 2453 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  W )
)
226proplem3 14743 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
2321, 5, 22grppropd 15670 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  ( W  e.  Grp  <->  A  e.  Grp ) )
2420, 23mpbid 210 . 2  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  Grp )
25183ad2ant1 1009 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  W  e.  Ring )
26 inss2 3674 . . . . 5  |-  ( S  i^i  ( Base `  W
) )  C_  ( Base `  W )
2726sseli 3455 . . . 4  |-  ( x  e.  ( S  i^i  ( Base `  W )
)  ->  x  e.  ( Base `  W )
)
28273ad2ant2 1010 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  x  e.  ( Base `  W )
)
29 simp3 990 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  y  e.  ( Base `  W )
)
303, 13rngcl 16776 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3125, 28, 29, 30syl3anc 1219 . 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 994 . . . 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 3457 . . 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 995 . . 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 996 . . 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 16781 . . 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 1221 . 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 994 . . . 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 3457 . . 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 995 . . . 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 3457 . . 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 996 . . 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 16782 . . 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 1221 . 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 16779 . . 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 1221 . 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 16786 . . 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 17072 1  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3430   ` cfv 5521  (class class class)co 6195   Basecbs 14287   ↾s cress 14288   +g cplusg 14352   .rcmulr 14353   Grpcgrp 15524   1rcur 16720   Ringcrg 16763  SubRingcsubrg 16979   LModclmod 17066  subringAlg csra 17367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-ip 14370  df-0g 14494  df-mnd 15529  df-grp 15659  df-subg 15792  df-mgp 16709  df-ur 16721  df-rng 16765  df-subrg 16981  df-lmod 17068  df-sra 17371
This theorem is referenced by:  rlmlmod  17404  sraassa  17514  sranlm  20392
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