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Theorem aspval 21646
Description: Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
aspval.a 𝐴 = (AlgSpanβ€˜π‘Š)
aspval.v 𝑉 = (Baseβ€˜π‘Š)
aspval.l 𝐿 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
aspval ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
Distinct variable groups:   𝑑,𝐿   𝑑,𝑆   𝑑,𝑉   𝑑,π‘Š
Allowed substitution hint:   𝐴(𝑑)

Proof of Theorem aspval
Dummy variables 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aspval.a . . . . 5 𝐴 = (AlgSpanβ€˜π‘Š)
2 fveq2 6890 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
3 aspval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
42, 3eqtr4di 2788 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
54pweqd 4618 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
6 fveq2 6890 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘Š))
7 fveq2 6890 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 aspval.l . . . . . . . . . . 11 𝐿 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2788 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝐿)
106, 9ineq12d 4212 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) = ((SubRingβ€˜π‘Š) ∩ 𝐿))
1110rabeqdv 3445 . . . . . . . 8 (𝑀 = π‘Š β†’ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
1211inteqd 4954 . . . . . . 7 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
135, 12mpteq12dv 5238 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
14 df-asp 21628 . . . . . 6 AlgSpan = (𝑀 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑}))
153fvexi 6904 . . . . . . . 8 𝑉 ∈ V
1615pwex 5377 . . . . . . 7 𝒫 𝑉 ∈ V
1716mptex 7226 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}) ∈ V
1813, 14, 17fvmpt 6997 . . . . 5 (π‘Š ∈ AssAlg β†’ (AlgSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
191, 18eqtrid 2782 . . . 4 (π‘Š ∈ AssAlg β†’ 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
2019fveq1d 6892 . . 3 (π‘Š ∈ AssAlg β†’ (π΄β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})β€˜π‘†))
2120adantr 479 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})β€˜π‘†))
22 eqid 2730 . . 3 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
23 sseq1 4006 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑑))
2423rabbidv 3438 . . . 4 (𝑠 = 𝑆 β†’ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
2524inteqd 4954 . . 3 (𝑠 = 𝑆 β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
26 simpr 483 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
2715elpw2 5344 . . . 4 (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 βŠ† 𝑉)
2826, 27sylibr 233 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 ∈ 𝒫 𝑉)
29 assaring 21635 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
303subrgid 20463 . . . . . . 7 (π‘Š ∈ Ring β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
3129, 30syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
32 assalmod 21634 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
333, 8lss1 20693 . . . . . . 7 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝐿)
3432, 33syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ 𝐿)
3531, 34elind 4193 . . . . 5 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿))
36 sseq2 4007 . . . . . 6 (𝑑 = 𝑉 β†’ (𝑆 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑉))
3736rspcev 3611 . . . . 5 ((𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
3835, 37sylan 578 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
39 intexrab 5339 . . . 4 (βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑 ↔ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑} ∈ V)
4038, 39sylib 217 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑} ∈ V)
4122, 25, 28, 40fvmptd3 7020 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})β€˜π‘†) = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
4221, 41eqtrd 2770 1 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  {crab 3430  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆ© cint 4949   ↦ cmpt 5230  β€˜cfv 6542  Basecbs 17148  Ringcrg 20127  SubRingcsubrg 20457  LModclmod 20614  LSubSpclss 20686  AssAlgcasa 21624  AlgSpancasp 21625
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-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-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-mgp 20029  df-ur 20076  df-ring 20129  df-subrg 20459  df-lmod 20616  df-lss 20687  df-assa 21627  df-asp 21628
This theorem is referenced by:  asplss  21647  aspid  21648  aspsubrg  21649  aspss  21650  aspssid  21651  aspval2  21671
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