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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 | ⊢ 𝑁 = (RSpan‘𝑅) |
elrsp.b | ⊢ 𝐵 = (Base‘𝑅) |
elrsp.1 | ⊢ 0 = (0g‘𝑅) |
elrsp.x | ⊢ · = (.r‘𝑅) |
elrsp.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
elrsp.i | ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
Ref | Expression |
---|---|
elrsp | ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐼) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝐼)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrsp.n | . . . 4 ⊢ 𝑁 = (RSpan‘𝑅) | |
2 | rspval 20018 | . . . 4 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2782 | . . 3 ⊢ 𝑁 = (LSpan‘(ringLMod‘𝑅)) |
4 | elrsp.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
5 | rlmbas 20020 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
6 | 4, 5 | eqtri 2782 | . . 3 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
7 | eqid 2759 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
8 | eqid 2759 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
9 | eqid 2759 | . . 3 ⊢ (0g‘(Scalar‘(ringLMod‘𝑅))) = (0g‘(Scalar‘(ringLMod‘𝑅))) | |
10 | elrsp.x | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | rlmvsca 20027 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | 10, 11 | eqtri 2782 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
13 | elrsp.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
14 | rlmlmod 20030 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
16 | elrsp.i | . . 3 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) | |
17 | 3, 6, 7, 8, 9, 12, 15, 16 | ellspds 31070 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐼) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘(ringLMod‘𝑅))) ↑m 𝐼)(𝑎 finSupp (0g‘(Scalar‘(ringLMod‘𝑅))) ∧ 𝑋 = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) |
18 | rlmsca 20025 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
19 | 13, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
20 | 19 | fveq2d 6655 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
21 | 4, 20 | syl5eq 2806 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
22 | 21 | oveq1d 7158 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐼) = ((Base‘(Scalar‘(ringLMod‘𝑅))) ↑m 𝐼)) |
23 | elrsp.1 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
24 | 19 | fveq2d 6655 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘(ringLMod‘𝑅)))) |
25 | 23, 24 | syl5eq 2806 | . . . . 5 ⊢ (𝜑 → 0 = (0g‘(Scalar‘(ringLMod‘𝑅)))) |
26 | 25 | breq2d 5037 | . . . 4 ⊢ (𝜑 → (𝑎 finSupp 0 ↔ 𝑎 finSupp (0g‘(Scalar‘(ringLMod‘𝑅))))) |
27 | 4 | fvexi 6665 | . . . . . . . . 9 ⊢ 𝐵 ∈ V |
28 | 27 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V) |
29 | 28, 16 | ssexd 5187 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V) |
30 | 29 | mptexd 6971 | . . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)) ∈ V) |
31 | 5 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(ringLMod‘𝑅))) |
32 | rlmplusg 20021 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = (+g‘(ringLMod‘𝑅))) |
34 | 30, 13, 15, 31, 33 | gsumpropd 17939 | . . . . 5 ⊢ (𝜑 → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖))) = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))) |
35 | 34 | eqeq2d 2770 | . . . 4 ⊢ (𝜑 → (𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖))) ↔ 𝑋 = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖))))) |
36 | 26, 35 | anbi12d 634 | . . 3 ⊢ (𝜑 → ((𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))) ↔ (𝑎 finSupp (0g‘(Scalar‘(ringLMod‘𝑅))) ∧ 𝑋 = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) |
37 | 22, 36 | rexeqbidv 3318 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝐵 ↑m 𝐼)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘(ringLMod‘𝑅))) ↑m 𝐼)(𝑎 finSupp (0g‘(Scalar‘(ringLMod‘𝑅))) ∧ 𝑋 = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) |
38 | 17, 37 | bitr4d 285 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝐼) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝐼)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 Vcvv 3407 ⊆ wss 3854 class class class wbr 5025 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 ↑m cmap 8409 finSupp cfsupp 8851 Basecbs 16526 +gcplusg 16608 .rcmulr 16609 Scalarcsca 16611 ·𝑠 cvsca 16612 0gc0g 16756 Σg cgsu 16757 Ringcrg 19350 LModclmod 19687 LSpanclspn 19796 ringLModcrglmod 19994 RSpancrsp 19996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-sup 8924 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-fzo 13068 df-seq 13404 df-hash 13726 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-ds 16630 df-hom 16632 df-cco 16633 df-0g 16758 df-gsum 16759 df-prds 16764 df-pws 16766 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-mhm 18007 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-mulg 18277 df-subg 18328 df-ghm 18408 df-cntz 18499 df-cmn 18960 df-abl 18961 df-mgp 19293 df-ur 19305 df-ring 19352 df-subrg 19586 df-lmod 19689 df-lss 19757 df-lsp 19797 df-lmhm 19847 df-lbs 19900 df-sra 19997 df-rgmod 19998 df-rsp 20000 df-nzr 20084 df-dsmm 20482 df-frlm 20497 df-uvc 20533 |
This theorem is referenced by: elrspunidl 31112 |
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