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Theorem sralmod 15939
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 10 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  A  =  ( ( subringAlg  `  W ) `
 S ) )
3 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
43subrgss 15546 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  S  C_  ( Base `  W ) )
52, 4srabase 15931 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  A )
)
62, 4sraaddg 15932 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  A ) )
72, 4srasca 15934 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  =  (Scalar `  A ) )
82, 4sravsca 15935 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .s `  A ) )
9 eqid 2283 . . 3  |-  ( Ws  S )  =  ( Ws  S )
109, 3ressbas 13198 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( S  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  S
) ) )
11 eqid 2283 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
129, 11ressplusg 13250 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( +g  `  W )  =  ( +g  `  ( Ws  S ) ) )
13 eqid 2283 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
149, 13ressmulr 13261 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( .r `  W )  =  ( .r `  ( Ws  S ) ) )
15 eqid 2283 . . 3  |-  ( 1r
`  W )  =  ( 1r `  W
)
169, 15subrg1 15555 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( 1r `  W )  =  ( 1r `  ( Ws  S ) ) )
179subrgrng 15548 . 2  |-  ( S  e.  (SubRing `  W
)  ->  ( Ws  S
)  e.  Ring )
18 subrgrcl 15550 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Ring )
19 rnggrp 15346 . . . 4  |-  ( W  e.  Ring  ->  W  e. 
Grp )
2018, 19syl 15 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  W  e.  Grp )
21 eqidd 2284 . . . 4  |-  ( S  e.  (SubRing `  W
)  ->  ( Base `  W )  =  (
Base `  W )
)
226proplem3 13593 . . . 4  |-  ( ( S  e.  (SubRing `  W
)  /\  ( x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
) )  ->  (
x ( +g  `  W
) y )  =  ( x ( +g  `  A ) y ) )
2321, 5, 22grppropd 14500 . . 3  |-  ( S  e.  (SubRing `  W
)  ->  ( W  e.  Grp  <->  A  e.  Grp ) )
2420, 23mpbid 201 . 2  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  Grp )
25183ad2ant1 976 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  W  e.  Ring )
26 inss2 3390 . . . . 5  |-  ( S  i^i  ( Base `  W
) )  C_  ( Base `  W )
2726sseli 3176 . . . 4  |-  ( x  e.  ( S  i^i  ( Base `  W )
)  ->  x  e.  ( Base `  W )
)
28273ad2ant2 977 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  x  e.  ( Base `  W )
)
29 simp3 957 . . 3  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  y  e.  ( Base `  W )
)
303, 13rngcl 15354 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3125, 28, 29, 30syl3anc 1182 . 2  |-  ( ( S  e.  (SubRing `  W
)  /\  x  e.  ( S  i^i  ( Base `  W ) )  /\  y  e.  (
Base `  W )
)  ->  ( x
( .r `  W
) y )  e.  ( Base `  W
) )
3218adantr 451 . . 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 961 . . . 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 3178 . . 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 962 . . 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 963 . . 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 15359 . . 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 1184 . 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 451 . . 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 961 . . . 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 3178 . . 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 962 . . . 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 3178 . . 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 963 . . 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 15360 . . 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 1184 . 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 15357 . . 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 1184 . 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 15364 . . 3  |-  ( ( W  e.  Ring  /\  x  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) x )  =  x )
5018, 49sylan 457 . 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 15633 1  |-  ( S  e.  (SubRing `  W
)  ->  A  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   Grpcgrp 14362   Ringcrg 15337   1rcur 15339  SubRingcsubrg 15541   LModclmod 15627   subringAlg csra 15921
This theorem is referenced by:  rlmlmod  15957  sraassa  16065  sranlm  18195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-sra 15925
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