Theorem issubassa 19786

Description: The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
issubassa.s	⊢ 𝑆 = (𝑊 ↾s 𝐴)
issubassa.l	⊢ 𝐿 = (LSubSp‘𝑊)
issubassa.v	⊢ 𝑉 = (Base‘𝑊)
issubassa.o	⊢ 1 = (1r‘𝑊)

Assertion

Ref	Expression
issubassa	⊢ ((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)))

Proof of Theorem issubassa

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

Step	Hyp	Ref	Expression
1		simpl1 1184	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ AssAlg)
2		assaring 19782	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
3	1, 2	syl 17	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ Ring)
4		issubassa.s	. . . . 5 ⊢ 𝑆 = (𝑊 ↾s 𝐴)
5		assaring 19782	. . . . . 6 ⊢ (𝑆 ∈ AssAlg → 𝑆 ∈ Ring)
6	5	adantl 482	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ Ring)
7	4, 6	syl5eqelr 2888	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊 ↾s 𝐴) ∈ Ring)
8		simpl3 1186	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ⊆ 𝑉)
9		simpl2 1185	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 1 ∈ 𝐴)
10	8, 9	jca 512	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ⊆ 𝑉 ∧ 1 ∈ 𝐴))
11		issubassa.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
12		issubassa.o	. . . . 5 ⊢ 1 = (1r‘𝑊)
13	11, 12	issubrg 19225	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑊) ↔ ((𝑊 ∈ Ring ∧ (𝑊 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝑉 ∧ 1 ∈ 𝐴)))
14	3, 7, 10, 13	syl21anbrc 1337	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ (SubRing‘𝑊))
15		assalmod 19781	. . . . 5 ⊢ (𝑆 ∈ AssAlg → 𝑆 ∈ LMod)
16	15	adantl 482	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ LMod)
17		assalmod 19781	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
18		issubassa.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
19	4, 11, 18	islss3 19421	. . . . 5 ⊢ (𝑊 ∈ LMod → (𝐴 ∈ 𝐿 ↔ (𝐴 ⊆ 𝑉 ∧ 𝑆 ∈ LMod)))
20	1, 17, 19	3syl 18	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ 𝐿 ↔ (𝐴 ⊆ 𝑉 ∧ 𝑆 ∈ LMod)))
21	8, 16, 20	mpbir2and 709	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ 𝐿)
22	14, 21	jca 512	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿))
23	11	subrgss 19226	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑊) → 𝐴 ⊆ 𝑉)
24	23	ad2antrl 724	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝐴 ⊆ 𝑉)
25	4, 11	ressbas2 16384	. . . . 5 ⊢ (𝐴 ⊆ 𝑉 → 𝐴 = (Base‘𝑆))
26	24, 25	syl 17	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝐴 = (Base‘𝑆))
27		eqid 2795	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	4, 27	resssca 16479	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑆))
29	28	ad2antrl 724	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
30		eqidd 2796	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
31		eqid 2795	. . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
32	4, 31	ressvsca 16480	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑊) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑆))
33	32	ad2antrl 724	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑆))
34		eqid 2795	. . . . . 6 ⊢ (.r‘𝑊) = (.r‘𝑊)
35	4, 34	ressmulr 16454	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑊) → (.r‘𝑊) = (.r‘𝑆))
36	35	ad2antrl 724	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (.r‘𝑊) = (.r‘𝑆))
37		simpr 485	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿) → 𝐴 ∈ 𝐿)
38	4, 18	lsslmod 19422	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐿) → 𝑆 ∈ LMod)
39	17, 37, 38	syl2an 595	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ LMod)
40	4	subrgring 19228	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
41	40	ad2antrl 724	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ Ring)
42	27	assasca 19783	. . . . 5 ⊢ (𝑊 ∈ AssAlg → (Scalar‘𝑊) ∈ CRing)
43	42	adantr 481	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (Scalar‘𝑊) ∈ CRing)
44		simpll 763	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑊 ∈ AssAlg)
45		simpr1 1187	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
46	24	adantr 481	. . . . . 6 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐴 ⊆ 𝑉)
47		simpr2 1188	. . . . . 6 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
48	46, 47	sseldd 3890	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝑉)
49		simpr3 1189	. . . . . 6 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
50	46, 49	sseldd 3890	. . . . 5 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝑉)
51		eqid 2795	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
52	11, 27, 51, 31, 34	assaass 19779	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
53	44, 45, 48, 50, 52	syl13anc 1365	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
54	11, 27, 51, 31, 34	assaassr 19780	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(.r‘𝑊)(𝑥( ·𝑠 ‘𝑊)𝑧)) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
55	44, 45, 48, 50, 54	syl13anc 1365	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(.r‘𝑊)(𝑥( ·𝑠 ‘𝑊)𝑧)) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
56	26, 29, 30, 33, 36, 39, 41, 43, 53, 55	isassad 19785	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)
57	56	3ad2antl1 1178	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)
58	22, 57	impbida 797	1 ⊢ ((𝑊 ∈ AssAlg ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ⊆ wss 3859  ‘cfv 6225  (class class class)co 7016  Basecbs 16312   ↾_s cress 16313  ._rcmulr 16395  Scalarcsca 16397   ·_𝑠 cvsca 16398  1_rcur 18941  Ringcrg 18987  CRingccrg 18988  SubRingcsubrg 19221  LModclmod 19324  LSubSpclss 19393  AssAlgcasa 19771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866  df-subg 18030  df-mgp 18930  df-ur 18942  df-ring 18989  df-subrg 19223  df-lmod 19326  df-lss 19394  df-assa 19774

This theorem is referenced by:  rnasclassa  19811  mplassa  19922  ply1assa  20050