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Theorem aspval2 21451
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 3964 . . . . . . . . 9 (π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ↔ (π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)))
21anbi1i 624 . . . . . . . 8 ((π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯) ↔ ((π‘₯ ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (LSubSpβ€˜π‘Š)) ∧ 𝑆 βŠ† π‘₯))
3 anass 469 . . . . . . . 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 274 . . . . . . 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 2732 . . . . . . . . . . 11 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
75, 6issubassa2 21445 . . . . . . . . . 10 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ (π‘₯ ∈ (LSubSpβ€˜π‘Š) ↔ ran 𝐢 βŠ† π‘₯))
87anbi1d 630 . . . . . . . . 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 4184 . . . . . . . . 9 ((ran 𝐢 βŠ† π‘₯ ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯)
108, 9bitrdi 286 . . . . . . . 8 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (SubRingβ€˜π‘Š)) β†’ ((π‘₯ ∈ (LSubSpβ€˜π‘Š) ∧ 𝑆 βŠ† π‘₯) ↔ (ran 𝐢 βˆͺ 𝑆) βŠ† π‘₯))
1110pm5.32da 579 . . . . . . 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 282 . . . . . 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 2801 . . . . 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 481 . . . 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 3433 . . . 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 3433 . . . 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 2797 . . 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 4955 . 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 21426 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯})
22 assaring 21415 . . . 4 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
2320subrgmre 20343 . . . 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 2732 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
26 assalmod 21414 . . . . . . 7 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
27 eqid 2732 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
285, 25, 22, 26, 27, 20asclf 21435 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝐢:(Baseβ€˜(Scalarβ€˜π‘Š))βŸΆπ‘‰)
2928frnd 6725 . . . . 5 (π‘Š ∈ AssAlg β†’ ran 𝐢 βŠ† 𝑉)
3029adantr 481 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ ran 𝐢 βŠ† 𝑉)
31 simpr 485 . . . 4 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
3230, 31unssd 4186 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (ran 𝐢 βˆͺ 𝑆) βŠ† 𝑉)
33 aspval2.r . . . 4 𝑅 = (mrClsβ€˜(SubRingβ€˜π‘Š))
3433mrcval 17553 . . 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 683 . 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 2782 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  {crab 3432   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ© cint 4950  ran crn 5677  β€˜cfv 6543  Basecbs 17143  Scalarcsca 17199  Moorecmre 17525  mrClscmrc 17526  Ringcrg 20055  SubRingcsubrg 20314  LSubSpclss 20541  AssAlgcasa 21404  AlgSpancasp 21405  algSccascl 21406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-0g 17386  df-mre 17529  df-mrc 17530  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-mgp 19987  df-ur 20004  df-ring 20057  df-subrg 20316  df-lmod 20472  df-lss 20542  df-lsp 20582  df-assa 21407  df-asp 21408  df-ascl 21409
This theorem is referenced by:  evlseu  21645
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