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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 31094 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
aspval.a | β’ π΄ = (AlgSpanβπ) |
aspval.v | β’ π = (Baseβπ) |
aspval.l | β’ πΏ = (LSubSpβπ) |
Ref | Expression |
---|---|
aspid | β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β (π΄βπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β π β AssAlg) | |
2 | aspval.v | . . . . 5 β’ π = (Baseβπ) | |
3 | 2 | subrgss 20470 | . . . 4 β’ (π β (SubRingβπ) β π β π) |
4 | 3 | 3ad2ant2 1131 | . . 3 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β π β π) |
5 | aspval.a | . . . 4 β’ π΄ = (AlgSpanβπ) | |
6 | aspval.l | . . . 4 β’ πΏ = (LSubSpβπ) | |
7 | 5, 2, 6 | aspval 21756 | . . 3 β’ ((π β AssAlg β§ π β π) β (π΄βπ) = β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘}) |
8 | 1, 4, 7 | syl2anc 583 | . 2 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β (π΄βπ) = β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘}) |
9 | 3simpc 1147 | . . . 4 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β (π β (SubRingβπ) β§ π β πΏ)) | |
10 | elin 3957 | . . . 4 β’ (π β ((SubRingβπ) β© πΏ) β (π β (SubRingβπ) β§ π β πΏ)) | |
11 | 9, 10 | sylibr 233 | . . 3 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β π β ((SubRingβπ) β© πΏ)) |
12 | intmin 4963 | . . 3 β’ (π β ((SubRingβπ) β© πΏ) β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} = π) | |
13 | 11, 12 | syl 17 | . 2 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β β© {π‘ β ((SubRingβπ) β© πΏ) β£ π β π‘} = π) |
14 | 8, 13 | eqtrd 2764 | 1 β’ ((π β AssAlg β§ π β (SubRingβπ) β§ π β πΏ) β (π΄βπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3424 β© cin 3940 β wss 3941 β© cint 4941 βcfv 6534 Basecbs 17149 SubRingcsubrg 20465 LSubSpclss 20774 AssAlgcasa 21734 AlgSpancasp 21735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 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-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-mgp 20036 df-ur 20083 df-ring 20136 df-subrg 20467 df-lmod 20704 df-lss 20775 df-assa 21737 df-asp 21738 |
This theorem is referenced by: mplbas2 21928 mplind 21962 |
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