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Theorem aspval2 19168
 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 ∧ 𝑆𝑉) → (𝐴𝑆) = (𝑅‘(ran 𝐶𝑆)))

Proof of Theorem aspval2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3758 . . . . . . . . 9 (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
21anbi1i 727 . . . . . . . 8 ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆𝑥))
3 anass 679 . . . . . . . 8 (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)))
42, 3bitri 263 . . . . . . 7 ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)))
5 aspval2.c . . . . . . . . . . 11 𝐶 = (algSc‘𝑊)
6 eqid 2610 . . . . . . . . . . 11 (LSubSp‘𝑊) = (LSubSp‘𝑊)
75, 6issubassa2 19166 . . . . . . . . . 10 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶𝑥))
87anbi1d 737 . . . . . . . . 9 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥) ↔ (ran 𝐶𝑥𝑆𝑥)))
9 unss 3749 . . . . . . . . 9 ((ran 𝐶𝑥𝑆𝑥) ↔ (ran 𝐶𝑆) ⊆ 𝑥)
108, 9syl6bb 275 . . . . . . . 8 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥) ↔ (ran 𝐶𝑆) ⊆ 𝑥))
1110pm5.32da 671 . . . . . . 7 (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)))
124, 11syl5bb 271 . . . . . 6 (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)))
1312abbidv 2728 . . . . 5 (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)})
1413adantr 480 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)})
15 df-rab 2905 . . . 4 {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)}
16 df-rab 2905 . . . 4 {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)}
1714, 15, 163eqtr4g 2669 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
1817inteqd 4415 . 2 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
19 aspval2.a . . 3 𝐴 = (AlgSpan‘𝑊)
20 aspval2.v . . 3 𝑉 = (Base‘𝑊)
2119, 20, 6aspval 19149 . 2 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥})
22 assaring 19141 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
2320subrgmre 18627 . . . . 5 (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
2422, 23syl 17 . . . 4 (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
2524adantr 480 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (SubRing‘𝑊) ∈ (Moore‘𝑉))
26 eqid 2610 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
27 assalmod 19140 . . . . . . 7 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
28 eqid 2610 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
295, 26, 22, 27, 28, 20asclf 19158 . . . . . 6 (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
30 frn 5966 . . . . . 6 (𝐶:(Base‘(Scalar‘𝑊))⟶𝑉 → ran 𝐶𝑉)
3129, 30syl 17 . . . . 5 (𝑊 ∈ AssAlg → ran 𝐶𝑉)
3231adantr 480 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → ran 𝐶𝑉)
33 simpr 476 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆𝑉)
3432, 33unssd 3751 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (ran 𝐶𝑆) ⊆ 𝑉)
35 aspval2.r . . . 4 𝑅 = (mrCls‘(SubRing‘𝑊))
3635mrcval 16093 . . 3 (((SubRing‘𝑊) ∈ (Moore‘𝑉) ∧ (ran 𝐶𝑆) ⊆ 𝑉) → (𝑅‘(ran 𝐶𝑆)) = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
3725, 34, 36syl2anc 691 . 2 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝑅‘(ran 𝐶𝑆)) = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
3818, 21, 373eqtr4d 2654 1 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = (𝑅‘(ran 𝐶𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∩ cint 4410  ran crn 5039  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  Scalarcsca 15771  Moorecmre 16065  mrClscmrc 16066  Ringcrg 18370  SubRingcsubrg 18599  LSubSpclss 18753  AssAlgcasa 19130  AlgSpancasp 19131  algSccascl 19132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mre 16069  df-mrc 16070  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-assa 19133  df-asp 19134  df-ascl 19135 This theorem is referenced by:  evlseu  19337
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