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Theorem aspval 21793
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 6891 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
3 aspval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
42, 3eqtr4di 2785 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
54pweqd 4615 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
6 fveq2 6891 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘Š))
7 fveq2 6891 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 aspval.l . . . . . . . . . . 11 𝐿 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2785 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝐿)
106, 9ineq12d 4209 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) = ((SubRingβ€˜π‘Š) ∩ 𝐿))
1110rabeqdv 3442 . . . . . . . 8 (𝑀 = π‘Š β†’ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
1211inteqd 4949 . . . . . . 7 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
135, 12mpteq12dv 5233 . . . . . 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 21775 . . . . . 6 AlgSpan = (𝑀 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑}))
153fvexi 6905 . . . . . . . 8 𝑉 ∈ V
1615pwex 5374 . . . . . . 7 𝒫 𝑉 ∈ V
1716mptex 7229 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}) ∈ V
1813, 14, 17fvmpt 6999 . . . . 5 (π‘Š ∈ AssAlg β†’ (AlgSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
191, 18eqtrid 2779 . . . 4 (π‘Š ∈ AssAlg β†’ 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
2019fveq1d 6893 . . 3 (π‘Š ∈ AssAlg β†’ (π΄β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})β€˜π‘†))
2120adantr 480 . 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 2727 . . 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 4003 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑑))
2423rabbidv 3435 . . . 4 (𝑠 = 𝑆 β†’ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
2524inteqd 4949 . . 3 (𝑠 = 𝑆 β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
26 simpr 484 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
2715elpw2 5341 . . . 4 (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 βŠ† 𝑉)
2826, 27sylibr 233 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 ∈ 𝒫 𝑉)
29 assaring 21782 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
303subrgid 20501 . . . . . . 7 (π‘Š ∈ Ring β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
3129, 30syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
32 assalmod 21781 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
333, 8lss1 20811 . . . . . . 7 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝐿)
3432, 33syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ 𝐿)
3531, 34elind 4190 . . . . 5 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿))
36 sseq2 4004 . . . . . 6 (𝑑 = 𝑉 β†’ (𝑆 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑉))
3736rspcev 3607 . . . . 5 ((𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
3835, 37sylan 579 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
39 intexrab 5336 . . . 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 7022 . 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 2767 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 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065  {crab 3427  Vcvv 3469   ∩ cin 3943   βŠ† wss 3944  π’« cpw 4598  βˆ© cint 4944   ↦ cmpt 5225  β€˜cfv 6542  Basecbs 17171  Ringcrg 20164  SubRingcsubrg 20495  LModclmod 20732  LSubSpclss 20804  AssAlgcasa 21771  AlgSpancasp 21772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-0g 17414  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-grp 18884  df-mgp 20066  df-ur 20113  df-ring 20166  df-subrg 20497  df-lmod 20734  df-lss 20805  df-assa 21774  df-asp 21775
This theorem is referenced by:  asplss  21794  aspid  21795  aspsubrg  21796  aspss  21797  aspssid  21798  aspval2  21818
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