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Theorem aspval2 19562
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 3947 . . . . . . . . 9 (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)))
21anbi1i 610 . . . . . . . 8 ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ ((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆𝑥))
3 anass 454 . . . . . . . 8 (((𝑥 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)))
42, 3bitri 264 . . . . . . 7 ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)))
5 aspval2.c . . . . . . . . . . 11 𝐶 = (algSc‘𝑊)
6 eqid 2771 . . . . . . . . . . 11 (LSubSp‘𝑊) = (LSubSp‘𝑊)
75, 6issubassa2 19560 . . . . . . . . . 10 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → (𝑥 ∈ (LSubSp‘𝑊) ↔ ran 𝐶𝑥))
87anbi1d 615 . . . . . . . . 9 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥) ↔ (ran 𝐶𝑥𝑆𝑥)))
9 unss 3938 . . . . . . . . 9 ((ran 𝐶𝑥𝑆𝑥) ↔ (ran 𝐶𝑆) ⊆ 𝑥)
108, 9syl6bb 276 . . . . . . . 8 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (SubRing‘𝑊)) → ((𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥) ↔ (ran 𝐶𝑆) ⊆ 𝑥))
1110pm5.32da 568 . . . . . . 7 (𝑊 ∈ AssAlg → ((𝑥 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (LSubSp‘𝑊) ∧ 𝑆𝑥)) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)))
124, 11syl5bb 272 . . . . . 6 (𝑊 ∈ AssAlg → ((𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥) ↔ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)))
1312abbidv 2890 . . . . 5 (𝑊 ∈ AssAlg → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)})
1413adantr 466 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)})
15 df-rab 3070 . . . 4 {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥} = {𝑥 ∣ (𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∧ 𝑆𝑥)}
16 df-rab 3070 . . . 4 {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ (SubRing‘𝑊) ∧ (ran 𝐶𝑆) ⊆ 𝑥)}
1714, 15, 163eqtr4g 2830 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥} = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
1817inteqd 4617 . 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 19543 . 2 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = {𝑥 ∈ ((SubRing‘𝑊) ∩ (LSubSp‘𝑊)) ∣ 𝑆𝑥})
22 assaring 19535 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
2320subrgmre 19014 . . . . 5 (𝑊 ∈ Ring → (SubRing‘𝑊) ∈ (Moore‘𝑉))
2422, 23syl 17 . . . 4 (𝑊 ∈ AssAlg → (SubRing‘𝑊) ∈ (Moore‘𝑉))
2524adantr 466 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (SubRing‘𝑊) ∈ (Moore‘𝑉))
26 eqid 2771 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
27 assalmod 19534 . . . . . . 7 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
28 eqid 2771 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
295, 26, 22, 27, 28, 20asclf 19552 . . . . . 6 (𝑊 ∈ AssAlg → 𝐶:(Base‘(Scalar‘𝑊))⟶𝑉)
3029frnd 6191 . . . . 5 (𝑊 ∈ AssAlg → ran 𝐶𝑉)
3130adantr 466 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → ran 𝐶𝑉)
32 simpr 471 . . . 4 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆𝑉)
3331, 32unssd 3940 . . 3 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (ran 𝐶𝑆) ⊆ 𝑉)
34 aspval2.r . . . 4 𝑅 = (mrCls‘(SubRing‘𝑊))
3534mrcval 16478 . . 3 (((SubRing‘𝑊) ∈ (Moore‘𝑉) ∧ (ran 𝐶𝑆) ⊆ 𝑉) → (𝑅‘(ran 𝐶𝑆)) = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
3625, 33, 35syl2anc 573 . 2 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝑅‘(ran 𝐶𝑆)) = {𝑥 ∈ (SubRing‘𝑊) ∣ (ran 𝐶𝑆) ⊆ 𝑥})
3718, 21, 363eqtr4d 2815 1 ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = (𝑅‘(ran 𝐶𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  {crab 3065  cun 3721  cin 3722  wss 3723   cint 4612  ran crn 5251  cfv 6030  Basecbs 16064  Scalarcsca 16152  Moorecmre 16450  mrClscmrc 16451  Ringcrg 18755  SubRingcsubrg 18986  LSubSpclss 19142  AssAlgcasa 19524  AlgSpancasp 19525  algSccascl 19526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-0g 16310  df-mre 16454  df-mrc 16455  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-mgp 18698  df-ur 18710  df-ring 18757  df-subrg 18988  df-lmod 19075  df-lss 19143  df-lsp 19185  df-assa 19527  df-asp 19528  df-ascl 19529
This theorem is referenced by:  evlseu  19731
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