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Mirrors > Home > MPE Home > Th. List > rnascl | Structured version Visualization version GIF version |
Description: The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
rnascl.a | β’ π΄ = (algScβπ) |
rnascl.o | β’ 1 = (1rβπ) |
rnascl.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
rnascl | β’ (π β AssAlg β ran π΄ = (πβ{ 1 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnascl.a | . . . 4 β’ π΄ = (algScβπ) | |
2 | eqid 2728 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
3 | eqid 2728 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
4 | eqid 2728 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
5 | rnascl.o | . . . 4 β’ 1 = (1rβπ) | |
6 | 1, 2, 3, 4, 5 | asclfval 21812 | . . 3 β’ π΄ = (π¦ β (Baseβ(Scalarβπ)) β¦ (π¦( Β·π βπ) 1 )) |
7 | 6 | rnmpt 5957 | . 2 β’ ran π΄ = {π₯ β£ βπ¦ β (Baseβ(Scalarβπ))π₯ = (π¦( Β·π βπ) 1 )} |
8 | assalmod 21794 | . . 3 β’ (π β AssAlg β π β LMod) | |
9 | assaring 21795 | . . . 4 β’ (π β AssAlg β π β Ring) | |
10 | eqid 2728 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
11 | 10, 5 | ringidcl 20202 | . . . 4 β’ (π β Ring β 1 β (Baseβπ)) |
12 | 9, 11 | syl 17 | . . 3 β’ (π β AssAlg β 1 β (Baseβπ)) |
13 | rnascl.n | . . . 4 β’ π = (LSpanβπ) | |
14 | 2, 3, 10, 4, 13 | lspsn 20886 | . . 3 β’ ((π β LMod β§ 1 β (Baseβπ)) β (πβ{ 1 }) = {π₯ β£ βπ¦ β (Baseβ(Scalarβπ))π₯ = (π¦( Β·π βπ) 1 )}) |
15 | 8, 12, 14 | syl2anc 583 | . 2 β’ (π β AssAlg β (πβ{ 1 }) = {π₯ β£ βπ¦ β (Baseβ(Scalarβπ))π₯ = (π¦( Β·π βπ) 1 )}) |
16 | 7, 15 | eqtr4id 2787 | 1 β’ (π β AssAlg β ran π΄ = (πβ{ 1 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {cab 2705 βwrex 3067 {csn 4629 ran crn 5679 βcfv 6548 (class class class)co 7420 Basecbs 17180 Scalarcsca 17236 Β·π cvsca 17237 1rcur 20121 Ringcrg 20173 LModclmod 20743 LSpanclspn 20855 AssAlgcasa 21784 algSccascl 21786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-lss 20816 df-lsp 20856 df-assa 21787 df-ascl 21789 |
This theorem is referenced by: issubassa2 21825 |
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