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Mirrors > Home > MPE Home > Th. List > aspid | Structured version Visualization version GIF version |
Description: The algebraic span of a subalgebra is itself. (spanid 28757 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
aspval.a | ⊢ 𝐴 = (AlgSpan‘𝑊) |
aspval.v | ⊢ 𝑉 = (Base‘𝑊) |
aspval.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
aspid | ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1170 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ AssAlg) | |
2 | aspval.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 2 | subrgss 19144 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ 𝑉) |
4 | 3 | 3ad2ant2 1168 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → 𝑆 ⊆ 𝑉) |
5 | aspval.a | . . . 4 ⊢ 𝐴 = (AlgSpan‘𝑊) | |
6 | aspval.l | . . . 4 ⊢ 𝐿 = (LSubSp‘𝑊) | |
7 | 5, 2, 6 | aspval 19696 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡}) |
8 | 1, 4, 7 | syl2anc 579 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡}) |
9 | 3simpc 1186 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿)) | |
10 | elin 4025 | . . . 4 ⊢ (𝑆 ∈ ((SubRing‘𝑊) ∩ 𝐿) ↔ (𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿)) | |
11 | 9, 10 | sylibr 226 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ ((SubRing‘𝑊) ∩ 𝐿)) |
12 | intmin 4719 | . . 3 ⊢ (𝑆 ∈ ((SubRing‘𝑊) ∩ 𝐿) → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} = 𝑆) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡} = 𝑆) |
14 | 8, 13 | eqtrd 2861 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 {crab 3121 ∩ cin 3797 ⊆ wss 3798 ∩ cint 4699 ‘cfv 6127 Basecbs 16229 SubRingcsubrg 19139 LSubSpclss 19295 AssAlgcasa 19677 AlgSpancasp 19678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-mgp 18851 df-ur 18863 df-ring 18910 df-subrg 19141 df-lmod 19228 df-lss 19296 df-assa 19680 df-asp 19681 |
This theorem is referenced by: mplbas2 19838 mplind 19869 |
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