Theorem sraassa 20093

Description: The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)

Hypothesis

Ref	Expression
sraassa.a	⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)

Assertion

Ref	Expression
sraassa	⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Proof of Theorem sraassa

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

Step	Hyp	Ref	Expression
1		sraassa.a	. . . 4 ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
2	1	a1i 11	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3		eqid 2821	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 19530	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	4	adantl 484	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
6	2, 5	srabase 19944	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
7	2, 5	srasca 19947	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8		eqid 2821	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
9	8	subrgbas 19538	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
10	9	adantl 484	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
11	2, 5	sravsca 19948	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
12	2, 5	sramulr 19946	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = (.r‘𝐴))
13	1	sralmod 19953	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
14	13	adantl 484	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15		crngring 19302	. . . 4 ⊢ (𝑊 ∈ CRing → 𝑊 ∈ Ring)
16	15	adantr 483	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17		eqidd 2822	. . . 4 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
18	2, 5	sraaddg 19945	. . . . 5 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
19	18	oveqdr 7178	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
20	12	oveqdr 7178	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)𝑦) = (𝑥(.r‘𝐴)𝑦))
21	17, 6, 19, 20	ringpropd 19326	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ Ring ↔ 𝐴 ∈ Ring))
22	16, 21	mpbid 234	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ Ring)
23	8	subrgcrng 19533	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ CRing)
24	16	adantr 483	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
25	5	adantr 483	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑆 ⊆ (Base‘𝑊))
26		simpr1 1190	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ 𝑆)
27	25, 26	sseldd 3968	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
28		simpr2 1191	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
29		simpr3 1192	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
30		eqid 2821	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
31	3, 30	ringass 19308	. . 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)))
32	24, 27, 28, 29, 31	syl13anc 1368	. 2 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
33		eqid 2821	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
34	33	crngmgp 19299	. . . 4 ⊢ (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
35	34	ad2antrr 724	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
36	33, 3	mgpbas 19239	. . . 4 ⊢ (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
37	33, 30	mgpplusg 19237	. . . 4 ⊢ (.r‘𝑊) = (+g‘(mulGrp‘𝑊))
38	36, 37	cmn12 18921	. . 3 ⊢ (((mulGrp‘𝑊) ∈ CMnd ∧ (𝑦 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r‘𝑊)(𝑥(.r‘𝑊)𝑧)) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
39	35, 28, 27, 29, 38	syl13anc 1368	. 2 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r‘𝑊)(𝑥(.r‘𝑊)𝑧)) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
40	6, 7, 10, 11, 12, 14, 22, 23, 32, 39	isassad 20090	1 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ⊆ wss 3936  ‘cfv 6350  (class class class)co 7150  Basecbs 16477   ↾_s cress 16478  +_gcplusg 16559  ._rcmulr 16560  CMndccmn 18900  mulGrpcmgp 19233  Ringcrg 19291  CRingccrg 19292  SubRingcsubrg 19525  LModclmod 19628  subringAlg csra 19934  AssAlgcasa 20076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-subg 18270  df-cmn 18902  df-mgp 19234  df-ur 19246  df-ring 19293  df-cring 19294  df-subrg 19527  df-lmod 19630  df-sra 19938  df-assa 20079

This theorem is referenced by:  rlmassa  20094