Theorem aspval2 21936

Description: The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypotheses

Ref	Expression
aspval2.a	⊢ 𝐴 = (AlgSpan‘𝑊)
aspval2.c	⊢ 𝐶 = (algSc‘𝑊)
aspval2.r	⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))
aspval2.v	⊢ 𝑉 = (Base‘𝑊)

Assertion

Ref	Expression
aspval2	⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

Proof of Theorem aspval2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3979	. . . . . . . . 9 ⊢ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
2	1	anbi1i 624	. . . . . . . 8 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥))
3		anass 468	. . . . . . . 8 ⊢ (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
4	2, 3	bitri 275	. . . . . . 7 ⊢ ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)))
5		aspval2.c	. . . . . . . . . . 11 ⊢ 𝐶 = (algSc‘𝑊)
6		eqid 2735	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
7	5, 6	issubassa2 21930	. . . . . . . . . 10 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶 ⊆ 𝑥))
8	7	anbi1d 631	. . . . . . . . 9 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥)))
9		unss 4200	. . . . . . . . 9 ⊢ ((ran 𝐶 ⊆ 𝑥 ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)
10	8, 9	bitrdi 287	. . . . . . . 8 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥) ↔ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥))
11	10	pm5.32da 579	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆 ⊆ 𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
12	4, 11	bitrid 283	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)))
13	12	abbidv 2806	. . . . 5 ⊢ (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
14	13	adantr 480	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)})
15		df-rab 3434	. . . 4 ⊢ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆 ⊆ 𝑥)}
16		df-rab 3434	. . . 4 ⊢ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥)}
17	14, 15, 16	3eqtr4g 2800	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
18	17	inteqd 4956	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥} = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
19		aspval2.a	. . 3 ⊢ 𝐴 = (AlgSpan‘𝑊)
20		aspval2.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
21	19, 20, 6	aspval 21911	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆 ⊆ 𝑥})
22		assaring 21899	. . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
23	20	subrgmre 20614	. . . 4 ⊢ (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
24	22, 23	syl 17	. . 3 ⊢ (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
25		eqid 2735	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
26		assalmod 21898	. . . . . . 7 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
27		eqid 2735	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
28	5, 25, 22, 26, 27, 20	asclf 21920	. . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
29	28	frnd 6745	. . . . 5 ⊢ (𝑊 ∈ AssAlg → ran 𝐶 ⊆ 𝑉)
30	29	adantr 480	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → ran 𝐶 ⊆ 𝑉)
31		simpr 484	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
32	30, 31	unssd 4202	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (ran 𝐶 ∪ 𝑆) ⊆ 𝑉)
33		aspval2.r	. . . 4 ⊢ 𝑅 = (mrCls‘(SubRing‘𝑊))
34	33	mrcval 17655	. . 3 ⊢ (((SubRing‘𝑊) ∈ (Moore‘𝑉) ∧ (ran 𝐶 ∪ 𝑆) ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
35	24, 32, 34	syl2an2r 685	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝑅‘(ran 𝐶 ∪ 𝑆)) = ∩ {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶 ∪ 𝑆) ⊆ 𝑥})
36	18, 21, 35	3eqtr4d 2785	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = (𝑅‘(ran 𝐶 ∪ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  {crab 3433   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∩ cint 4951  ran crn 5690  ‘cfv 6563  Basecbs 17245  Scalarcsca 17301  Moorecmre 17627  mrClscmrc 17628  Ringcrg 20251  SubRingcsubrg 20586  LSubSpclss 20947  AssAlgcasa 21888  AlgSpancasp 21889  algSccascl 21890

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mre 17631  df-mrc 17632  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-assa 21891  df-asp 21892  df-ascl 21893

This theorem is referenced by:  evlseu  22125