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Mirrors > Home > MPE Home > Th. List > ellspd | Structured version Visualization version GIF version |
Description: The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
Ref | Expression |
---|---|
ellspd.n | ⊢ 𝑁 = (LSpan‘𝑀) |
ellspd.v | ⊢ 𝐵 = (Base‘𝑀) |
ellspd.k | ⊢ 𝐾 = (Base‘𝑆) |
ellspd.s | ⊢ 𝑆 = (Scalar‘𝑀) |
ellspd.z | ⊢ 0 = (0g‘𝑆) |
ellspd.t | ⊢ · = ( ·𝑠 ‘𝑀) |
ellspd.f | ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
ellspd.m | ⊢ (𝜑 → 𝑀 ∈ LMod) |
ellspd.i | ⊢ (𝜑 → 𝐼 ∈ V) |
Ref | Expression |
---|---|
ellspd | ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellspd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
2 | ffn 6508 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼) | |
3 | fnima 6472 | . . . . . 6 ⊢ (𝐹 Fn 𝐼 → (𝐹 “ 𝐼) = ran 𝐹) | |
4 | 1, 2, 3 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐼) = ran 𝐹) |
5 | 4 | fveq2d 6668 | . . . 4 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = (𝑁‘ran 𝐹)) |
6 | eqid 2821 | . . . . . 6 ⊢ (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) | |
7 | 6 | rnmpt 5821 | . . . . 5 ⊢ ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} |
8 | eqid 2821 | . . . . . 6 ⊢ (𝑆 freeLMod 𝐼) = (𝑆 freeLMod 𝐼) | |
9 | eqid 2821 | . . . . . 6 ⊢ (Base‘(𝑆 freeLMod 𝐼)) = (Base‘(𝑆 freeLMod 𝐼)) | |
10 | ellspd.v | . . . . . 6 ⊢ 𝐵 = (Base‘𝑀) | |
11 | ellspd.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑀) | |
12 | ellspd.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ LMod) | |
13 | ellspd.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) | |
14 | ellspd.s | . . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑀) | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑀)) |
16 | ellspd.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑀) | |
17 | 8, 9, 10, 11, 6, 12, 13, 15, 1, 16 | frlmup3 20938 | . . . . 5 ⊢ (𝜑 → ran (𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼)) ↦ (𝑀 Σg (𝑓 ∘f · 𝐹))) = (𝑁‘ran 𝐹)) |
18 | 7, 17 | syl5eqr 2870 | . . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} = (𝑁‘ran 𝐹)) |
19 | 5, 18 | eqtr4d 2859 | . . 3 ⊢ (𝜑 → (𝑁‘(𝐹 “ 𝐼)) = {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))}) |
20 | 19 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ 𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))})) |
21 | ovex 7183 | . . . . . 6 ⊢ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V | |
22 | eleq1 2900 | . . . . . 6 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → (𝑋 ∈ V ↔ (𝑀 Σg (𝑓 ∘f · 𝐹)) ∈ V)) | |
23 | 21, 22 | mpbiri 260 | . . . . 5 ⊢ (𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
24 | 23 | rexlimivw 3282 | . . . 4 ⊢ (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) → 𝑋 ∈ V) |
25 | eqeq1 2825 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) | |
26 | 25 | rexbidv 3297 | . . . 4 ⊢ (𝑎 = 𝑋 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
27 | 24, 26 | elab3 3673 | . . 3 ⊢ (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} ↔ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))) |
28 | 14 | fvexi 6678 | . . . . . . 7 ⊢ 𝑆 ∈ V |
29 | ellspd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝑆) | |
30 | ellspd.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑆) | |
31 | eqid 2821 | . . . . . . . 8 ⊢ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } | |
32 | 8, 29, 30, 31 | frlmbas 20893 | . . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐼 ∈ V) → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
33 | 28, 13, 32 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 } = (Base‘(𝑆 freeLMod 𝐼))) |
34 | 33 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → (Base‘(𝑆 freeLMod 𝐼)) = {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }) |
35 | 34 | rexeqdv 3416 | . . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
36 | breq1 5061 | . . . . 5 ⊢ (𝑎 = 𝑓 → (𝑎 finSupp 0 ↔ 𝑓 finSupp 0 )) | |
37 | 36 | rexrab 3686 | . . . 4 ⊢ (∃𝑓 ∈ {𝑎 ∈ (𝐾 ↑m 𝐼) ∣ 𝑎 finSupp 0 }𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) |
38 | 35, 37 | syl6bb 289 | . . 3 ⊢ (𝜑 → (∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
39 | 27, 38 | syl5bb 285 | . 2 ⊢ (𝜑 → (𝑋 ∈ {𝑎 ∣ ∃𝑓 ∈ (Base‘(𝑆 freeLMod 𝐼))𝑎 = (𝑀 Σg (𝑓 ∘f · 𝐹))} ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
40 | 20, 39 | bitrd 281 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 {crab 3142 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 ran crn 5550 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 ↑m cmap 8400 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Σg cgsu 16708 LModclmod 19628 LSpanclspn 19737 freeLMod cfrlm 20884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lmhm 19788 df-lbs 19841 df-sra 19938 df-rgmod 19939 df-nzr 20025 df-dsmm 20870 df-frlm 20885 df-uvc 20921 |
This theorem is referenced by: elfilspd 20941 islindf4 20976 ellspds 30928 fedgmul 31022 |
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