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Theorem aspval 21427
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 6892 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
3 aspval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
42, 3eqtr4di 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
54pweqd 4620 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
6 fveq2 6892 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘Š))
7 fveq2 6892 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 aspval.l . . . . . . . . . . 11 𝐿 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2791 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝐿)
106, 9ineq12d 4214 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) = ((SubRingβ€˜π‘Š) ∩ 𝐿))
1110rabeqdv 3448 . . . . . . . 8 (𝑀 = π‘Š β†’ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
1211inteqd 4956 . . . . . . 7 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
135, 12mpteq12dv 5240 . . . . . 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 21409 . . . . . 6 AlgSpan = (𝑀 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑}))
153fvexi 6906 . . . . . . . 8 𝑉 ∈ V
1615pwex 5379 . . . . . . 7 𝒫 𝑉 ∈ V
1716mptex 7225 . . . . . 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 2785 . . . 4 (π‘Š ∈ AssAlg β†’ 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
2019fveq1d 6894 . . 3 (π‘Š ∈ AssAlg β†’ (π΄β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})β€˜π‘†))
2120adantr 482 . 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 2733 . . 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 4008 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑑))
2423rabbidv 3441 . . . 4 (𝑠 = 𝑆 β†’ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
2524inteqd 4956 . . 3 (𝑠 = 𝑆 β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
26 simpr 486 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
2715elpw2 5346 . . . 4 (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 βŠ† 𝑉)
2826, 27sylibr 233 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 ∈ 𝒫 𝑉)
29 assaring 21416 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
303subrgid 20321 . . . . . . 7 (π‘Š ∈ Ring β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
3129, 30syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
32 assalmod 21415 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
333, 8lss1 20549 . . . . . . 7 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝐿)
3432, 33syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ 𝐿)
3531, 34elind 4195 . . . . 5 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿))
36 sseq2 4009 . . . . . 6 (𝑑 = 𝑉 β†’ (𝑆 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑉))
3736rspcev 3613 . . . . 5 ((𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
3835, 37sylan 581 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿)𝑆 βŠ† 𝑑)
39 intexrab 5341 . . . 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 2773 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆ© cint 4951   ↦ cmpt 5232  β€˜cfv 6544  Basecbs 17144  Ringcrg 20056  SubRingcsubrg 20315  LModclmod 20471  LSubSpclss 20542  AssAlgcasa 21405  AlgSpancasp 21406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-mgp 19988  df-ur 20005  df-ring 20058  df-subrg 20317  df-lmod 20473  df-lss 20543  df-assa 21408  df-asp 21409
This theorem is referenced by:  asplss  21428  aspid  21429  aspsubrg  21430  aspss  21431  aspssid  21432  aspval2  21452
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