Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 20184 | . . . 4 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2759 | . . 3 ⊢ 𝑁 = (LSpan‘(ringLMod‘𝑅)) |
4 | elrsp.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
5 | rlmbas 20186 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
6 | 4, 5 | eqtri 2759 | . . 3 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
7 | eqid 2736 | . . 3 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
8 | eqid 2736 | . . 3 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
9 | eqid 2736 | . . 3 ⊢ (0g‘(Scalar‘(ringLMod‘𝑅))) = (0g‘(Scalar‘(ringLMod‘𝑅))) | |
10 | elrsp.x | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | rlmvsca 20193 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | 10, 11 | eqtri 2759 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
13 | elrsp.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
14 | rlmlmod 20196 | . . . 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 31232 | . 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 20191 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
19 | 13, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
20 | 19 | fveq2d 6699 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
21 | 4, 20 | syl5eq 2783 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
22 | 21 | oveq1d 7206 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐼) = ((Base‘(Scalar‘(ringLMod‘𝑅))) ↑m 𝐼)) |
23 | elrsp.1 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
24 | 19 | fveq2d 6699 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘(Scalar‘(ringLMod‘𝑅)))) |
25 | 23, 24 | syl5eq 2783 | . . . . 5 ⊢ (𝜑 → 0 = (0g‘(Scalar‘(ringLMod‘𝑅)))) |
26 | 25 | breq2d 5051 | . . . 4 ⊢ (𝜑 → (𝑎 finSupp 0 ↔ 𝑎 finSupp (0g‘(Scalar‘(ringLMod‘𝑅))))) |
27 | 4 | fvexi 6709 | . . . . . . . . 9 ⊢ 𝐵 ∈ V |
28 | 27 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V) |
29 | 28, 16 | ssexd 5202 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V) |
30 | 29 | mptexd 7018 | . . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)) ∈ V) |
31 | 5 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(ringLMod‘𝑅))) |
32 | rlmplusg 20187 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = (+g‘(ringLMod‘𝑅))) |
34 | 30, 13, 15, 31, 33 | gsumpropd 18104 | . . . . 5 ⊢ (𝜑 → (𝑅 Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖))) = ((ringLMod‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑎‘𝑖) · 𝑖)))) |
35 | 34 | eqeq2d 2747 | . . . 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 3304 | . 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 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ⊆ wss 3853 class class class wbr 5039 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 finSupp cfsupp 8963 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 Scalarcsca 16752 ·𝑠 cvsca 16753 0gc0g 16898 Σg cgsu 16899 Ringcrg 19516 LModclmod 19853 LSpanclspn 19962 ringLModcrglmod 20160 RSpancrsp 20162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lmhm 20013 df-lbs 20066 df-sra 20163 df-rgmod 20164 df-rsp 20166 df-nzr 20250 df-dsmm 20648 df-frlm 20663 df-uvc 20699 |
This theorem is referenced by: elrspunidl 31274 |
Copyright terms: Public domain | W3C validator |