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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 29118 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 1132 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ AssAlg) | |
2 | aspval.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 2 | subrgss 19530 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ 𝑉) |
4 | 3 | 3ad2ant2 1130 | . . 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 20096 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡}) |
8 | 1, 4, 7 | syl2anc 586 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = ∩ {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆 ⊆ 𝑡}) |
9 | 3simpc 1146 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿)) | |
10 | elin 4168 | . . . 4 ⊢ (𝑆 ∈ ((SubRing‘𝑊) ∩ 𝐿) ↔ (𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿)) | |
11 | 9, 10 | sylibr 236 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ ((SubRing‘𝑊) ∩ 𝐿)) |
12 | intmin 4888 | . . 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 2856 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆 ∈ 𝐿) → (𝐴‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ∩ cin 3934 ⊆ wss 3935 ∩ cint 4868 ‘cfv 6349 Basecbs 16477 SubRingcsubrg 19525 LSubSpclss 19697 AssAlgcasa 20076 AlgSpancasp 20077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ur 19246 df-ring 19293 df-subrg 19527 df-lmod 19630 df-lss 19698 df-assa 20079 df-asp 20080 |
This theorem is referenced by: mplbas2 20245 mplind 20276 |
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