Theorem sraassaOLD 21764

Description: Obsolete version of sraassa 21763 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 2726	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 20474	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	4	adantl 481	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
6	2, 5	srabase 21026	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
7	2, 5	srasca 21032	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8		eqid 2726	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
9	8	subrgbas 20483	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
10	9	adantl 481	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
11	2, 5	sravsca 21034	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
12	2, 5	sramulr 21030	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = (.r‘𝐴))
13	1	sralmod 21043	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
14	13	adantl 481	. 2 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15		crngring 20150	. . . 4 ⊢ (𝑊 ∈ CRing → 𝑊 ∈ Ring)
16	15	adantr 480	. . 3 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17		eqidd 2727	. . . 4 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
18	2, 5	sraaddg 21028	. . . . 5 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
19	18	oveqdr 7433	. . . 4 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
20	12	oveqdr 7433	. . . 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 20187	. . 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 480	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
24	5	adantr 480	. . . 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 3978	. . 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 2726	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
30	3, 29	ringass 20158	. . 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 2726	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
33	32	crngmgp 20146	. . . 4 ⊢ (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
34	33	ad2antrr 723	. . 3 ⊢ (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
35	32, 3	mgpbas 20045	. . . 4 ⊢ (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
36	32, 29	mgpplusg 20043	. . . 4 ⊢ (.r‘𝑊) = (+g‘(mulGrp‘𝑊))
37	35, 36	cmn12 19722	. . 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 21759	1 ⊢ ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3943  ‘cfv 6537  (class class class)co 7405  Basecbs 17153   ↾_s cress 17182  +_gcplusg 17206  ._rcmulr 17207  CMndccmn 19700  mulGrpcmgp 20039  Ringcrg 20138  CRingccrg 20139  SubRingcsubrg 20469  LModclmod 20706  subringAlg csra 21019  AssAlgcasa 21745

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-ip 17224  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-subg 19050  df-cmn 19702  df-mgp 20040  df-ur 20087  df-ring 20140  df-cring 20141  df-subrg 20471  df-lmod 20708  df-sra 21021  df-assa 21748

This theorem is referenced by: (None)