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Mirrors > Home > MPE Home > Th. List > frlmup3 | Structured version Visualization version GIF version |
Description: The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) |
Ref | Expression |
---|---|
frlmup.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlmup.b | ⊢ 𝐵 = (Base‘𝐹) |
frlmup.c | ⊢ 𝐶 = (Base‘𝑇) |
frlmup.v | ⊢ · = ( ·𝑠 ‘𝑇) |
frlmup.e | ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) |
frlmup.t | ⊢ (𝜑 → 𝑇 ∈ LMod) |
frlmup.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
frlmup.r | ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) |
frlmup.a | ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
frlmup.k | ⊢ 𝐾 = (LSpan‘𝑇) |
Ref | Expression |
---|---|
frlmup3 | ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmup.f | . . . 4 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | frlmup.b | . . . 4 ⊢ 𝐵 = (Base‘𝐹) | |
3 | frlmup.c | . . . 4 ⊢ 𝐶 = (Base‘𝑇) | |
4 | frlmup.v | . . . 4 ⊢ · = ( ·𝑠 ‘𝑇) | |
5 | frlmup.e | . . . 4 ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) | |
6 | frlmup.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
7 | frlmup.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
8 | frlmup.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) | |
9 | frlmup.a | . . . 4 ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | frlmup1 20941 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
11 | eqid 2821 | . . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
12 | 11 | lmodring 19641 | . . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
13 | 6, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
14 | 8, 13 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | eqid 2821 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
16 | 15, 1, 2 | uvcff 20934 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
17 | 14, 7, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
18 | 17 | frnd 6520 | . . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵) |
19 | eqid 2821 | . . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
20 | frlmup.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑇) | |
21 | 2, 19, 20 | lmhmlsp 19820 | . . 3 ⊢ ((𝐸 ∈ (𝐹 LMHom 𝑇) ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
22 | 10, 18, 21 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
23 | 2, 3 | lmhmf 19805 | . . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
25 | 24 | ffnd 6514 | . . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵) |
26 | fnima 6477 | . . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸) |
28 | eqid 2821 | . . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹) | |
29 | 1, 15, 28 | frlmlbs 20940 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
30 | 14, 7, 29 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
31 | 2, 28, 19 | lbssp 19850 | . . . . . 6 ⊢ (ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹) → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵) |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)) = 𝐵) |
33 | 32 | eqcomd 2827 | . . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) |
34 | 33 | imaeq2d 5928 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
35 | 27, 34 | eqtr3d 2858 | . 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
36 | imaco 6103 | . . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) | |
37 | 9 | ffnd 6514 | . . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼) |
38 | 17 | ffnd 6514 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼) |
39 | fnco 6464 | . . . . . . . 8 ⊢ ((𝐸 Fn 𝐵 ∧ (𝑅 unitVec 𝐼) Fn 𝐼 ∧ ran (𝑅 unitVec 𝐼) ⊆ 𝐵) → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼) | |
40 | 25, 38, 18, 39 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼) |
41 | fvco2 6757 | . . . . . . . . 9 ⊢ (((𝑅 unitVec 𝐼) Fn 𝐼 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) | |
42 | 38, 41 | sylan 582 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) |
43 | 6 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod) |
44 | 7 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋) |
45 | 8 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇)) |
46 | 9 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶) |
47 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼) | |
48 | 1, 2, 3, 4, 5, 43, 44, 45, 46, 47, 15 | frlmup2 20942 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢)) |
49 | 42, 48 | eqtr2d 2857 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢)) |
50 | 37, 40, 49 | eqfnfvd 6804 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼))) |
51 | 50 | imaeq1d 5927 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼)) |
52 | fnima 6477 | . . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴) | |
53 | 37, 52 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴) |
54 | 51, 53 | eqtr3d 2858 | . . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴) |
55 | fnima 6477 | . . . . . 6 ⊢ ((𝑅 unitVec 𝐼) Fn 𝐼 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼)) | |
56 | 38, 55 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑅 unitVec 𝐼) “ 𝐼) = ran (𝑅 unitVec 𝐼)) |
57 | 56 | imaeq2d 5928 | . . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
58 | 36, 54, 57 | 3eqtr3a 2880 | . . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
59 | 58 | fveq2d 6673 | . 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
60 | 22, 35, 59 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ↦ cmpt 5145 ran crn 5555 “ cima 5557 ∘ ccom 5558 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∘f cof 7406 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 Σg cgsu 16713 Ringcrg 19296 LModclmod 19633 LSpanclspn 19742 LMHom clmhm 19790 LBasisclbs 19845 freeLMod cfrlm 20889 unitVec cuvc 20925 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lmhm 19793 df-lbs 19846 df-sra 19943 df-rgmod 19944 df-nzr 20030 df-dsmm 20875 df-frlm 20890 df-uvc 20926 |
This theorem is referenced by: ellspd 20945 indlcim 20983 lnrfg 39717 |
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