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Theorem aspval2 21452
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 3965 . . . . . . . . 9 (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)))
21anbi1i 625 . . . . . . . 8 ((π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ ((π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯))
3 anass 470 . . . . . . . 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 275 . . . . . . 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 2733 . . . . . . . . . . 11 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
75, 6issubassa2 21446 . . . . . . . . . 10 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ↔ ran 𝐢 βŠ† π‘₯))
87anbi1d 631 . . . . . . . . 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 4185 . . . . . . . . 9 ((ran 𝐢 βŠ† π‘₯ ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)
108, 9bitrdi 287 . . . . . . . 8 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯))
1110pm5.32da 580 . . . . . . 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 283 . . . . . 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 2802 . . . . 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 482 . . . 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 3434 . . . 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 3434 . . . 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 2798 . . 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 4956 . 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 21427 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯})
22 assaring 21416 . . . 4 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
2320subrgmre 20344 . . . 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 2733 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
26 assalmod 21415 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
27 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
285, 25, 22, 26, 27, 20asclf 21436 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝐢:(Baseβ€˜(Scalarβ€˜π‘Š))βŸΆπ‘‰)
2928frnd 6726 . . . . 5 (π‘Š ∈ AssAlg β†’ ran 𝐢 βŠ† 𝑉)
3029adantr 482 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ran 𝐢 βŠ† 𝑉)
31 simpr 486 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
3230, 31unssd 4187 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (ran 𝐢 βˆͺ 𝑆) βŠ† 𝑉)
33 aspval2.r . . . 4 𝑅 = (mrClsβ€˜(SubRingβ€˜π‘Š))
3433mrcval 17554 . . 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 684 . 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 2783 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3433   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ© cint 4951  ran crn 5678  β€˜cfv 6544  Basecbs 17144  Scalarcsca 17200  Moorecmre 17526  mrClscmrc 17527  Ringcrg 20056  SubRingcsubrg 20315  LSubSpclss 20542  AssAlgcasa 21405  AlgSpancasp 21406  algSccascl 21407
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-1st 7975  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-3 12276  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mre 17530  df-mrc 17531  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-mgp 19988  df-ur 20005  df-ring 20058  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-assa 21408  df-asp 21409  df-ascl 21410
This theorem is referenced by:  evlseu  21646
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