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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrsp | Structured version Visualization version GIF version |
Description: Write the elements of a ring span as finite linear combinations. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
elrsp.n | β’ π = (RSpanβπ ) |
elrsp.b | β’ π΅ = (Baseβπ ) |
elrsp.1 | β’ 0 = (0gβπ ) |
elrsp.x | β’ Β· = (.rβπ ) |
elrsp.r | β’ (π β π β Ring) |
elrsp.i | β’ (π β πΌ β π΅) |
Ref | Expression |
---|---|
elrsp | β’ (π β (π β (πβπΌ) β βπ β (π΅ βm πΌ)(π finSupp 0 β§ π = (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrsp.n | . . . 4 β’ π = (RSpanβπ ) | |
2 | rspval 21107 | . . . 4 β’ (RSpanβπ ) = (LSpanβ(ringLModβπ )) | |
3 | 1, 2 | eqtri 2756 | . . 3 β’ π = (LSpanβ(ringLModβπ )) |
4 | elrsp.b | . . . 4 β’ π΅ = (Baseβπ ) | |
5 | rlmbas 21086 | . . . 4 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
6 | 4, 5 | eqtri 2756 | . . 3 β’ π΅ = (Baseβ(ringLModβπ )) |
7 | eqid 2728 | . . 3 β’ (Baseβ(Scalarβ(ringLModβπ ))) = (Baseβ(Scalarβ(ringLModβπ ))) | |
8 | eqid 2728 | . . 3 β’ (Scalarβ(ringLModβπ )) = (Scalarβ(ringLModβπ )) | |
9 | eqid 2728 | . . 3 β’ (0gβ(Scalarβ(ringLModβπ ))) = (0gβ(Scalarβ(ringLModβπ ))) | |
10 | elrsp.x | . . . 4 β’ Β· = (.rβπ ) | |
11 | rlmvsca 21093 | . . . 4 β’ (.rβπ ) = ( Β·π β(ringLModβπ )) | |
12 | 10, 11 | eqtri 2756 | . . 3 β’ Β· = ( Β·π β(ringLModβπ )) |
13 | elrsp.r | . . . 4 β’ (π β π β Ring) | |
14 | rlmlmod 21096 | . . . 4 β’ (π β Ring β (ringLModβπ ) β LMod) | |
15 | 13, 14 | syl 17 | . . 3 β’ (π β (ringLModβπ ) β LMod) |
16 | elrsp.i | . . 3 β’ (π β πΌ β π΅) | |
17 | 3, 6, 7, 8, 9, 12, 15, 16 | ellspds 33093 | . 2 β’ (π β (π β (πβπΌ) β βπ β ((Baseβ(Scalarβ(ringLModβπ ))) βm πΌ)(π finSupp (0gβ(Scalarβ(ringLModβπ ))) β§ π = ((ringLModβπ ) Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))))) |
18 | rlmsca 21091 | . . . . . . 7 β’ (π β Ring β π = (Scalarβ(ringLModβπ ))) | |
19 | 13, 18 | syl 17 | . . . . . 6 β’ (π β π = (Scalarβ(ringLModβπ ))) |
20 | 19 | fveq2d 6901 | . . . . 5 β’ (π β (Baseβπ ) = (Baseβ(Scalarβ(ringLModβπ )))) |
21 | 4, 20 | eqtrid 2780 | . . . 4 β’ (π β π΅ = (Baseβ(Scalarβ(ringLModβπ )))) |
22 | 21 | oveq1d 7435 | . . 3 β’ (π β (π΅ βm πΌ) = ((Baseβ(Scalarβ(ringLModβπ ))) βm πΌ)) |
23 | elrsp.1 | . . . . . 6 β’ 0 = (0gβπ ) | |
24 | 19 | fveq2d 6901 | . . . . . 6 β’ (π β (0gβπ ) = (0gβ(Scalarβ(ringLModβπ )))) |
25 | 23, 24 | eqtrid 2780 | . . . . 5 β’ (π β 0 = (0gβ(Scalarβ(ringLModβπ )))) |
26 | 25 | breq2d 5160 | . . . 4 β’ (π β (π finSupp 0 β π finSupp (0gβ(Scalarβ(ringLModβπ ))))) |
27 | 4 | fvexi 6911 | . . . . . . . . 9 β’ π΅ β V |
28 | 27 | a1i 11 | . . . . . . . 8 β’ (π β π΅ β V) |
29 | 28, 16 | ssexd 5324 | . . . . . . 7 β’ (π β πΌ β V) |
30 | 29 | mptexd 7236 | . . . . . 6 β’ (π β (π β πΌ β¦ ((πβπ) Β· π)) β V) |
31 | 5 | a1i 11 | . . . . . 6 β’ (π β (Baseβπ ) = (Baseβ(ringLModβπ ))) |
32 | rlmplusg 21087 | . . . . . . 7 β’ (+gβπ ) = (+gβ(ringLModβπ )) | |
33 | 32 | a1i 11 | . . . . . 6 β’ (π β (+gβπ ) = (+gβ(ringLModβπ ))) |
34 | 30, 13, 15, 31, 33 | gsumpropd 18638 | . . . . 5 β’ (π β (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π))) = ((ringLModβπ ) Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))) |
35 | 34 | eqeq2d 2739 | . . . 4 β’ (π β (π = (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π))) β π = ((ringLModβπ ) Ξ£g (π β πΌ β¦ ((πβπ) Β· π))))) |
36 | 26, 35 | anbi12d 631 | . . 3 β’ (π β ((π finSupp 0 β§ π = (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))) β (π finSupp (0gβ(Scalarβ(ringLModβπ ))) β§ π = ((ringLModβπ ) Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))))) |
37 | 22, 36 | rexeqbidv 3340 | . 2 β’ (π β (βπ β (π΅ βm πΌ)(π finSupp 0 β§ π = (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))) β βπ β ((Baseβ(Scalarβ(ringLModβπ ))) βm πΌ)(π finSupp (0gβ(Scalarβ(ringLModβπ ))) β§ π = ((ringLModβπ ) Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))))) |
38 | 17, 37 | bitr4d 282 | 1 β’ (π β (π β (πβπΌ) β βπ β (π΅ βm πΌ)(π finSupp 0 β§ π = (π Ξ£g (π β πΌ β¦ ((πβπ) Β· π)))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwrex 3067 Vcvv 3471 β wss 3947 class class class wbr 5148 β¦ cmpt 5231 βcfv 6548 (class class class)co 7420 βm cmap 8845 finSupp cfsupp 9386 Basecbs 17180 +gcplusg 17233 .rcmulr 17234 Scalarcsca 17236 Β·π cvsca 17237 0gc0g 17421 Ξ£g cgsu 17422 Ringcrg 20173 LModclmod 20743 LSpanclspn 20855 ringLModcrglmod 21057 RSpancrsp 21103 |
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-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 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-se 5634 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-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 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-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-nzr 20452 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lmhm 20907 df-lbs 20960 df-sra 21058 df-rgmod 21059 df-rsp 21105 df-dsmm 21666 df-frlm 21681 df-uvc 21717 |
This theorem is referenced by: elrspunidl 33157 elrspunsn 33158 |
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