Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21103 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
11 | eqid 2736 | . . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
12 | 11 | lmodring 20229 | . . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
13 | 6, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
14 | 8, 13 | eqeltrd 2837 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | eqid 2736 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
16 | 15, 1, 2 | uvcff 21096 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
17 | 14, 7, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
18 | 17 | frnd 6653 | . . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵) |
19 | eqid 2736 | . . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
20 | frlmup.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑇) | |
21 | 2, 19, 20 | lmhmlsp 20409 | . . 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 584 | . 2 ⊢ (𝜑 → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
23 | 2, 3 | lmhmf 20394 | . . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
25 | 24 | ffnd 6646 | . . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵) |
26 | fnima 6608 | . . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸) |
28 | eqid 2736 | . . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹) | |
29 | 1, 15, 28 | frlmlbs 21102 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
30 | 14, 7, 29 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
31 | 2, 28, 19 | lbssp 20439 | . . . . . 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 2742 | . . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) |
34 | 33 | imaeq2d 5993 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
35 | 27, 34 | eqtr3d 2778 | . 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
36 | imaco 6183 | . . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) | |
37 | 9 | ffnd 6646 | . . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼) |
38 | 17 | ffnd 6646 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼) |
39 | fnco 6595 | . . . . . . . 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 1370 | . . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼) |
41 | fvco2 6915 | . . . . . . . . 9 ⊢ (((𝑅 unitVec 𝐼) Fn 𝐼 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) | |
42 | 38, 41 | sylan 580 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) |
43 | 6 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod) |
44 | 7 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋) |
45 | 8 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇)) |
46 | 9 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶) |
47 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼) | |
48 | 1, 2, 3, 4, 5, 43, 44, 45, 46, 47, 15 | frlmup2 21104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢)) |
49 | 42, 48 | eqtr2d 2777 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢)) |
50 | 37, 40, 49 | eqfnfvd 6962 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼))) |
51 | 50 | imaeq1d 5992 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼)) |
52 | fnima 6608 | . . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴) | |
53 | 37, 52 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴) |
54 | 51, 53 | eqtr3d 2778 | . . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴) |
55 | fnima 6608 | . . . . . 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 5993 | . . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
58 | 36, 54, 57 | 3eqtr3a 2800 | . . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
59 | 58 | fveq2d 6823 | . 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
60 | 22, 35, 59 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ⊆ wss 3897 ↦ cmpt 5172 ran crn 5615 “ cima 5617 ∘ ccom 5618 Fn wfn 6468 ⟶wf 6469 ‘cfv 6473 (class class class)co 7329 ∘f cof 7585 Basecbs 17001 Scalarcsca 17054 ·𝑠 cvsca 17055 Σg cgsu 17240 Ringcrg 19870 LModclmod 20221 LSpanclspn 20331 LMHom clmhm 20379 LBasisclbs 20434 freeLMod cfrlm 21051 unitVec cuvc 21087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-fzo 13476 df-seq 13815 df-hash 14138 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-hom 17075 df-cco 17076 df-0g 17241 df-gsum 17242 df-prds 17247 df-pws 17249 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-ghm 18920 df-cntz 19011 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-subrg 20119 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lmhm 20382 df-lbs 20435 df-sra 20532 df-rgmod 20533 df-nzr 20627 df-dsmm 21037 df-frlm 21052 df-uvc 21088 |
This theorem is referenced by: ellspd 21107 indlcim 21145 lnrfg 41195 |
Copyright terms: Public domain | W3C validator |