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Theorem sralmod 17946
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 2382 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 17543 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
52, 4srabase 17937 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  A )
)
62, 4sraaddg 17938 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
72, 4srasca 17940 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
82, 4sravsca 17941 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .s `  A ) )
9 eqid 2382 . . 3  |-  ( Ws  S )  =  ( Ws  S )
109, 3ressbas 14691 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( S  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  S
) ) )
11 eqid 2382 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
129, 11ressplusg 14748 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  ( Ws  S ) ) )
13 eqid 2382 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
149, 13ressmulr 14759 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  ( Ws  S ) ) )
15 eqid 2382 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
169, 15subrg1 17552 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  =  ( 1r `  ( Ws  S ) ) )
179subrgring 17545 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  e.  Ring )
18 subrgrcl 17547 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Ring )
19 ringgrp 17316 . . . 4  |-  ( W  e.  Ring  ->  W  e. 
Grp )
2018, 19syl 16 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Grp )
21 eqidd 2383 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  W )
)
226oveqdr 6220 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
2321, 5, 22grppropd 16185 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  ( W  e.  Grp  <->  A  e.  Grp ) )
2420, 23mpbid 210 . 2  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  Grp )
25183ad2ant1 1015 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  W  e.  Ring )
26 inss2 3633 . . . . 5  |-  ( S  i^i  ( Base `  W
) )  C_  ( Base `  W )
2726sseli 3413 . . . 4  |-  ( x  e.  ( S  i^i  ( Base `  W )
)  ->  x  e.  ( Base `  W )
)
28273ad2ant2 1016 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  x  e.  ( Base `  W )
)
29 simp3 996 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  y  e.  ( Base `  W )
)
303, 13ringcl 17325 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3125, 28, 29, 30syl3anc 1226 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3218adantr 463 . . 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 1000 . . . 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 3415 . . 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 1001 . . 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 1002 . . 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 17330 . . 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 1228 . 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 463 . . 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 1000 . . . 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 3415 . . 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 1001 . . . 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 3415 . . 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 1002 . . 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 17331 . . 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 1228 . 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 17328 . . 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 1228 . 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 17335 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) x )  =  x )
5018, 49sylan 469 . 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 17631 1  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    i^i cin 3388   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703   Grpcgrp 16170   1rcur 17266   Ringcrg 17311  SubRingcsubrg 17538   LModclmod 17625  subringAlg csra 17927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-ip 14720  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-subg 16315  df-mgp 17255  df-ur 17267  df-ring 17313  df-subrg 17540  df-lmod 17627  df-sra 17931
This theorem is referenced by:  rlmlmod  17964  sraassa  18087  sranlm  21278
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