Theorem sralmod 19961

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 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Proof of Theorem sralmod

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sralmod.a	. . . 4 ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
2	1	a1i 11	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3		eqid 2823	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 19538	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	2, 4	srabase 19952	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
6	2, 4	sraaddg 19953	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝐴))
7	2, 4	srasca 19955	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8	2, 4	sravsca 19956	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
9		eqid 2823	. . 3 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
10	9, 3	ressbas 16556	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝑆)))
11		eqid 2823	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
12	9, 11	ressplusg 16614	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘(𝑊 ↾s 𝑆)))
13		eqid 2823	. . 3 ⊢ (.r‘𝑊) = (.r‘𝑊)
14	9, 13	ressmulr 16627	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = (.r‘(𝑊 ↾s 𝑆)))
15		eqid 2823	. . 3 ⊢ (1r‘𝑊) = (1r‘𝑊)
16	9, 15	subrg1 19547	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘(𝑊 ↾s 𝑆)))
17	9	subrgring 19540	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) ∈ Ring)
18		subrgrcl 19542	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19		ringgrp 19304	. . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ Grp)
20	18, 19	syl 17	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21		eqidd 2824	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
22	6	oveqdr 7186	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
23	21, 5, 22	grppropd 18120	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
24	20, 23	mpbid 234	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25	18	3ad2ant1 1129	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26		elinel2 4175	. . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
27	26	3ad2ant2 1130	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
28		simp3 1134	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
29	3, 13	ringcl 19313	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
30	25, 27, 28, 29	syl3anc 1367	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
31	18	adantr 483	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
32		simpr1 1190	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
33	32	elin2d 4178	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
34		simpr2 1191	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
35		simpr3 1192	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
36	3, 11, 13	ringdi 19318	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)(𝑦(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑊)𝑦)(+g‘𝑊)(𝑥(.r‘𝑊)𝑧)))
37	31, 33, 34, 35, 36	syl13anc 1368	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)(𝑦(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑊)𝑦)(+g‘𝑊)(𝑥(.r‘𝑊)𝑧)))
38	18	adantr 483	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
39		simpr1 1190	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
40	39	elin2d 4178	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simpr2 1191	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
42	41	elin2d 4178	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
43		simpr3 1192	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
44	3, 11, 13	ringdir 19319	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g‘𝑊)𝑦)(.r‘𝑊)𝑧) = ((𝑥(.r‘𝑊)𝑧)(+g‘𝑊)(𝑦(.r‘𝑊)𝑧)))
45	38, 40, 42, 43, 44	syl13anc 1368	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g‘𝑊)𝑦)(.r‘𝑊)𝑧) = ((𝑥(.r‘𝑊)𝑧)(+g‘𝑊)(𝑦(.r‘𝑊)𝑧)))
46	3, 13	ringass 19316	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
47	38, 40, 42, 43, 46	syl13anc 1368	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
48	3, 13, 15	ringlidm 19323	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
49	18, 48	sylan 582	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
50	5, 6, 7, 8, 10, 12, 14, 16, 17, 24, 30, 37, 45, 47, 49	islmodd 19642	1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3937  ‘cfv 6357  (class class class)co 7158  Basecbs 16485   ↾_s cress 16486  +_gcplusg 16567  ._rcmulr 16568  Grpcgrp 18105  1_rcur 19253  Ringcrg 19299  SubRingcsubrg 19533  LModclmod 19636  subringAlg csra 19942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-ip 16585  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-subg 18278  df-mgp 19242  df-ur 19254  df-ring 19301  df-subrg 19535  df-lmod 19638  df-sra 19946

This theorem is referenced by:  rlmlmod  19979  sraassa  20101  sranlm  23295  sralvec  30992