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Theorem aspval 21292
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 6847 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
3 aspval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
42, 3eqtr4di 2795 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
54pweqd 4582 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
6 fveq2 6847 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘Š))
7 fveq2 6847 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 aspval.l . . . . . . . . . . 11 𝐿 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2795 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝐿)
106, 9ineq12d 4178 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) = ((SubRingβ€˜π‘Š) ∩ 𝐿))
1110rabeqdv 3425 . . . . . . . 8 (𝑀 = π‘Š β†’ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
1211inteqd 4917 . . . . . . 7 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑})
135, 12mpteq12dv 5201 . . . . . 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 21276 . . . . . 6 AlgSpan = (𝑀 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘€) ∩ (LSubSpβ€˜π‘€)) ∣ 𝑠 βŠ† 𝑑}))
153fvexi 6861 . . . . . . . 8 𝑉 ∈ V
1615pwex 5340 . . . . . . 7 𝒫 𝑉 ∈ V
1716mptex 7178 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}) ∈ V
1813, 14, 17fvmpt 6953 . . . . 5 (π‘Š ∈ AssAlg β†’ (AlgSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
191, 18eqtrid 2789 . . . 4 (π‘Š ∈ AssAlg β†’ 𝐴 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑}))
2019fveq1d 6849 . . 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 2737 . . 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 3974 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑑))
2423rabbidv 3418 . . . 4 (𝑠 = 𝑆 β†’ {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿) ∣ 𝑆 βŠ† 𝑑})
2524inteqd 4917 . . 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 5307 . . . 4 (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 βŠ† 𝑉)
2826, 27sylibr 233 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 ∈ 𝒫 𝑉)
29 assaring 21283 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
303subrgid 20240 . . . . . . 7 (π‘Š ∈ Ring β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
3129, 30syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ (SubRingβ€˜π‘Š))
32 assalmod 21282 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
333, 8lss1 20415 . . . . . . 7 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝐿)
3432, 33syl 17 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ 𝐿)
3531, 34elind 4159 . . . . 5 (π‘Š ∈ AssAlg β†’ 𝑉 ∈ ((SubRingβ€˜π‘Š) ∩ 𝐿))
36 sseq2 3975 . . . . . 6 (𝑑 = 𝑉 β†’ (𝑆 βŠ† 𝑑 ↔ 𝑆 βŠ† 𝑉))
3736rspcev 3584 . . . . 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 5302 . . . 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 6976 . 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 2777 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 3074  {crab 3410  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆ© cint 4912   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  Ringcrg 19971  SubRingcsubrg 20234  LModclmod 20338  LSubSpclss 20408  AssAlgcasa 21272  AlgSpancasp 21273
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-mgp 19904  df-ur 19921  df-ring 19973  df-subrg 20236  df-lmod 20340  df-lss 20409  df-assa 21275  df-asp 21276
This theorem is referenced by:  asplss  21293  aspid  21294  aspsubrg  21295  aspss  21296  aspssid  21297  aspval2  21317
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