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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 21066 | . . . 4 β’ (RSpanβπ ) = (LSpanβ(ringLModβπ )) | |
3 | 1, 2 | eqtri 2752 | . . 3 β’ π = (LSpanβ(ringLModβπ )) |
4 | elrsp.b | . . . 4 β’ π΅ = (Baseβπ ) | |
5 | rlmbas 21045 | . . . 4 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
6 | 4, 5 | eqtri 2752 | . . 3 β’ π΅ = (Baseβ(ringLModβπ )) |
7 | eqid 2724 | . . 3 β’ (Baseβ(Scalarβ(ringLModβπ ))) = (Baseβ(Scalarβ(ringLModβπ ))) | |
8 | eqid 2724 | . . 3 β’ (Scalarβ(ringLModβπ )) = (Scalarβ(ringLModβπ )) | |
9 | eqid 2724 | . . 3 β’ (0gβ(Scalarβ(ringLModβπ ))) = (0gβ(Scalarβ(ringLModβπ ))) | |
10 | elrsp.x | . . . 4 β’ Β· = (.rβπ ) | |
11 | rlmvsca 21052 | . . . 4 β’ (.rβπ ) = ( Β·π β(ringLModβπ )) | |
12 | 10, 11 | eqtri 2752 | . . 3 β’ Β· = ( Β·π β(ringLModβπ )) |
13 | elrsp.r | . . . 4 β’ (π β π β Ring) | |
14 | rlmlmod 21055 | . . . 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 32977 | . 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 21050 | . . . . . . 7 β’ (π β Ring β π = (Scalarβ(ringLModβπ ))) | |
19 | 13, 18 | syl 17 | . . . . . 6 β’ (π β π = (Scalarβ(ringLModβπ ))) |
20 | 19 | fveq2d 6886 | . . . . 5 β’ (π β (Baseβπ ) = (Baseβ(Scalarβ(ringLModβπ )))) |
21 | 4, 20 | eqtrid 2776 | . . . 4 β’ (π β π΅ = (Baseβ(Scalarβ(ringLModβπ )))) |
22 | 21 | oveq1d 7417 | . . 3 β’ (π β (π΅ βm πΌ) = ((Baseβ(Scalarβ(ringLModβπ ))) βm πΌ)) |
23 | elrsp.1 | . . . . . 6 β’ 0 = (0gβπ ) | |
24 | 19 | fveq2d 6886 | . . . . . 6 β’ (π β (0gβπ ) = (0gβ(Scalarβ(ringLModβπ )))) |
25 | 23, 24 | eqtrid 2776 | . . . . 5 β’ (π β 0 = (0gβ(Scalarβ(ringLModβπ )))) |
26 | 25 | breq2d 5151 | . . . 4 β’ (π β (π finSupp 0 β π finSupp (0gβ(Scalarβ(ringLModβπ ))))) |
27 | 4 | fvexi 6896 | . . . . . . . . 9 β’ π΅ β V |
28 | 27 | a1i 11 | . . . . . . . 8 β’ (π β π΅ β V) |
29 | 28, 16 | ssexd 5315 | . . . . . . 7 β’ (π β πΌ β V) |
30 | 29 | mptexd 7218 | . . . . . 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 21046 | . . . . . . 7 β’ (+gβπ ) = (+gβ(ringLModβπ )) | |
33 | 32 | a1i 11 | . . . . . 6 β’ (π β (+gβπ ) = (+gβ(ringLModβπ ))) |
34 | 30, 13, 15, 31, 33 | gsumpropd 18607 | . . . . 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 2735 | . . . 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 630 | . . 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 3335 | . 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 1533 β wcel 2098 βwrex 3062 Vcvv 3466 β wss 3941 class class class wbr 5139 β¦ cmpt 5222 βcfv 6534 (class class class)co 7402 βm cmap 8817 finSupp cfsupp 9358 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 Ξ£g cgsu 17391 Ringcrg 20134 LModclmod 20702 LSpanclspn 20814 ringLModcrglmod 21016 RSpancrsp 21062 |
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-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 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-se 5623 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-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 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-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-nzr 20411 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lmhm 20866 df-lbs 20919 df-sra 21017 df-rgmod 21018 df-rsp 21064 df-dsmm 21616 df-frlm 21631 df-uvc 21667 |
This theorem is referenced by: elrspunidl 33042 elrspunsn 33043 |
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