Theorem sraassaOLD 21807

Description: Obsolete version of sraassa 21806 as of 21-Mar-2025. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem sraassaOLD

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 2725	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 20515	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	4	adantl 480	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
6	2, 5	srabase 21067	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
7	2, 5	srasca 21073	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8		eqid 2725	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
9	8	subrgbas 20524	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
10	9	adantl 480	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
11	2, 5	sravsca 21075	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
12	2, 5	sramulr 21071	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = (.r‘𝐴))
13	1	sralmod 21084	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
14	13	adantl 480	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15		crngring 20189	. . . 4 ⊢ (𝑊 ∈ CRing → 𝑊 ∈ Ring)
16	15	adantr 479	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17		eqidd 2726	. . . 4 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
18	2, 5	sraaddg 21069	. . . . 5 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
19	18	oveqdr 7444	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
20	12	oveqdr 7444	. . . 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 20228	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ Ring ↔ 𝐴 ∈ Ring))
22	16, 21	mpbid 231	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ Ring)
23	16	adantr 479	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
24	5	adantr 479	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑆 ⊆ (Base‘𝑊))
25		simpr1 1191	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ 𝑆)
26	24, 25	sseldd 3973	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
27		simpr2 1192	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28		simpr3 1193	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
29		eqid 2725	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
30	3, 29	ringass 20197	. . 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)))
31	23, 26, 27, 28, 30	syl13anc 1369	. 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)))
32		eqid 2725	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
33	32	crngmgp 20185	. . . 4 ⊢ (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
34	33	ad2antrr 724	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
35	32, 3	mgpbas 20084	. . . 4 ⊢ (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
36	32, 29	mgpplusg 20082	. . . 4 ⊢ (.r‘𝑊) = (+g‘(mulGrp‘𝑊))
37	35, 36	cmn12 19761	. . 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)))
38	34, 27, 26, 28, 37	syl13anc 1369	. 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)))
39	6, 7, 10, 11, 12, 14, 22, 31, 38	isassad 21802	1 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3939  ‘cfv 6543  (class class class)co 7416  Basecbs 17179   ↾_s cress 17208  +_gcplusg 17232  ._rcmulr 17233  CMndccmn 19739  mulGrpcmgp 20078  Ringcrg 20177  CRingccrg 20178  SubRingcsubrg 20510  LModclmod 20747  subringAlg csra 21060  AssAlgcasa 21788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-ip 17250  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-subg 19082  df-cmn 19741  df-mgp 20079  df-ur 20126  df-ring 20179  df-cring 20180  df-subrg 20512  df-lmod 20749  df-sra 21062  df-assa 21791

This theorem is referenced by: (None)