Theorem sraassa 19325

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 2622	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 18781	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	4	adantl 482	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
6	2, 5	srabase 19178	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
7	2, 5	srasca 19181	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8		eqid 2622	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
9	8	subrgbas 18789	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
10	9	adantl 482	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
11	2, 5	sravsca 19182	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
12	2, 5	sramulr 19180	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = (.r‘𝐴))
13	1	sralmod 19187	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
14	13	adantl 482	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15		crngring 18558	. . . 4 ⊢ (𝑊 ∈ CRing → 𝑊 ∈ Ring)
16	15	adantr 481	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17		eqidd 2623	. . . 4 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
18	2, 5	sraaddg 19179	. . . . 5 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
19	18	oveqdr 6674	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
20	12	oveqdr 6674	. . . 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 18582	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ Ring ↔ 𝐴 ∈ Ring))
22	16, 21	mpbid 222	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ Ring)
23	8	subrgcrng 18784	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ CRing)
24	16	adantr 481	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
25	5	adantr 481	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑆 ⊆ (Base‘𝑊))
26		simpr1 1067	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ 𝑆)
27	25, 26	sseldd 3604	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
28		simpr2 1068	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
29		simpr3 1069	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
30		eqid 2622	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
31	3, 30	ringass 18564	. . 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 1328	. 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 2622	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
34	33	crngmgp 18555	. . . 4 ⊢ (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
35	34	ad2antrr 762	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
36	33, 3	mgpbas 18495	. . . 4 ⊢ (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
37	33, 30	mgpplusg 18493	. . . 4 ⊢ (.r‘𝑊) = (+g‘(mulGrp‘𝑊))
38	36, 37	cmn12 18213	. . 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 1328	. 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 19323	1 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Basecbs 15857   ↾_s cress 15858  +_gcplusg 15941  ._rcmulr 15942  CMndccmn 18193  mulGrpcmgp 18489  Ringcrg 18547  CRingccrg 18548  SubRingcsubrg 18776  LModclmod 18863  subringAlg csra 19168  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-sra 19172  df-assa 19312

This theorem is referenced by:  rlmassa  19326