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Theorem aspval2 21671
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 = (AlgSpanβ€˜π‘Š)
aspval2.c 𝐢 = (algScβ€˜π‘Š)
aspval2.r 𝑅 = (mrClsβ€˜(SubRingβ€˜π‘Š))
aspval2.v 𝑉 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
aspval2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = (π‘…β€˜(ran 𝐢 βˆͺ 𝑆)))

Proof of Theorem aspval2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 elin 3963 . . . . . . . . 9 (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)))
21anbi1i 622 . . . . . . . 8 ((π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ ((π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯))
3 anass 467 . . . . . . . 8 (((π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯)))
42, 3bitri 274 . . . . . . 7 ((π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯)))
5 aspval2.c . . . . . . . . . . 11 𝐢 = (algScβ€˜π‘Š)
6 eqid 2730 . . . . . . . . . . 11 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
75, 6issubassa2 21665 . . . . . . . . . 10 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ↔ ran 𝐢 βŠ† π‘₯))
87anbi1d 628 . . . . . . . . 9 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βŠ† π‘₯ ∧ 𝑆 βŠ† π‘₯)))
9 unss 4183 . . . . . . . . 9 ((ran 𝐢 βŠ† π‘₯ ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)
108, 9bitrdi 286 . . . . . . . 8 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯))
1110pm5.32da 577 . . . . . . 7 (π‘Š ∈ AssAlg β†’ ((π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯)) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)))
124, 11bitrid 282 . . . . . 6 (π‘Š ∈ AssAlg β†’ ((π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)))
1312abbidv 2799 . . . . 5 (π‘Š ∈ AssAlg β†’ {π‘₯ ∣ (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯)} = {π‘₯ ∣ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)})
1413adantr 479 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ {π‘₯ ∣ (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯)} = {π‘₯ ∣ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)})
15 df-rab 3431 . . . 4 {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯} = {π‘₯ ∣ (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯)}
16 df-rab 3431 . . . 4 {π‘₯ ∈ (SubRingβ€˜π‘Š) ∣ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯} = {π‘₯ ∣ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)}
1714, 15, 163eqtr4g 2795 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯} = {π‘₯ ∈ (SubRingβ€˜π‘Š) ∣ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯})
1817inteqd 4954 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ∩ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯} = ∩ {π‘₯ ∈ (SubRingβ€˜π‘Š) ∣ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯})
19 aspval2.a . . 3 𝐴 = (AlgSpanβ€˜π‘Š)
20 aspval2.v . . 3 𝑉 = (Baseβ€˜π‘Š)
2119, 20, 6aspval 21646 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯})
22 assaring 21635 . . . 4 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
2320subrgmre 20487 . . . 4 (π‘Š ∈ Ring β†’ (SubRingβ€˜π‘Š) ∈ (Mooreβ€˜π‘‰))
2422, 23syl 17 . . 3 (π‘Š ∈ AssAlg β†’ (SubRingβ€˜π‘Š) ∈ (Mooreβ€˜π‘‰))
25 eqid 2730 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
26 assalmod 21634 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
27 eqid 2730 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
285, 25, 22, 26, 27, 20asclf 21655 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝐢:(Baseβ€˜(Scalarβ€˜π‘Š))βŸΆπ‘‰)
2928frnd 6724 . . . . 5 (π‘Š ∈ AssAlg β†’ ran 𝐢 βŠ† 𝑉)
3029adantr 479 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ran 𝐢 βŠ† 𝑉)
31 simpr 483 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
3230, 31unssd 4185 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (ran 𝐢 βˆͺ 𝑆) βŠ† 𝑉)
33 aspval2.r . . . 4 𝑅 = (mrClsβ€˜(SubRingβ€˜π‘Š))
3433mrcval 17558 . . 3 (((SubRingβ€˜π‘Š) ∈ (Mooreβ€˜π‘‰) ∧ (ran 𝐢 βˆͺ 𝑆) βŠ† 𝑉) β†’ (π‘…β€˜(ran 𝐢 βˆͺ 𝑆)) = ∩ {π‘₯ ∈ (SubRingβ€˜π‘Š) ∣ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯})
3524, 32, 34syl2an2r 681 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π‘…β€˜(ran 𝐢 βˆͺ 𝑆)) = ∩ {π‘₯ ∈ (SubRingβ€˜π‘Š) ∣ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯})
3618, 21, 353eqtr4d 2780 1 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = (π‘…β€˜(ran 𝐢 βˆͺ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  {crab 3430   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ© cint 4949  ran crn 5676  β€˜cfv 6542  Basecbs 17148  Scalarcsca 17204  Moorecmre 17530  mrClscmrc 17531  Ringcrg 20127  SubRingcsubrg 20457  LSubSpclss 20686  AssAlgcasa 21624  AlgSpancasp 21625  algSccascl 21626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  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-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mre 17534  df-mrc 17535  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-subrng 20434  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-assa 21627  df-asp 21628  df-ascl 21629
This theorem is referenced by:  evlseu  21865
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