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Theorem aspval2 21317
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 3927 . . . . . . . . 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 21311 . . . . . . . . . 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 4145 . . . . . . . . 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 3407 . . . 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 3407 . . . 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 4913 . 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 21292 . 2 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (π΄β€˜π‘†) = ∩ {π‘₯ ∈ ((SubRingβ€˜π‘Š) ∩ (LSubSpβ€˜π‘Š)) ∣ 𝑆 βŠ† π‘₯})
22 assaring 21283 . . . 4 (π‘Š ∈ AssAlg β†’ π‘Š ∈ Ring)
2320subrgmre 20260 . . . 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 21282 . . . . . . 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 21301 . . . . . 6 (π‘Š ∈ AssAlg β†’ 𝐢:(Baseβ€˜(Scalarβ€˜π‘Š))βŸΆπ‘‰)
2928frnd 6677 . . . . 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 4147 . . 3 ((π‘Š ∈ AssAlg ∧ 𝑆 βŠ† 𝑉) β†’ (ran 𝐢 βˆͺ 𝑆) βŠ† 𝑉)
33 aspval2.r . . . 4 𝑅 = (mrClsβ€˜(SubRingβ€˜π‘Š))
3433mrcval 17495 . . 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 3406   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ© cint 4908  ran crn 5635  β€˜cfv 6497  Basecbs 17088  Scalarcsca 17141  Moorecmre 17467  mrClscmrc 17468  Ringcrg 19969  SubRingcsubrg 20232  LSubSpclss 20407  AssAlgcasa 21272  AlgSpancasp 21273  algSccascl 21274
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-0g 17328  df-mre 17471  df-mrc 17472  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-mgp 19902  df-ur 19919  df-ring 19971  df-subrg 20234  df-lmod 20338  df-lss 20408  df-lsp 20448  df-assa 21275  df-asp 21276  df-ascl 21277
This theorem is referenced by:  evlseu  21509
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