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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 21749 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
11 | eqid 2725 | . . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
12 | 11 | lmodring 20763 | . . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
13 | 6, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
14 | 8, 13 | eqeltrd 2825 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | eqid 2725 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
16 | 15, 1, 2 | uvcff 21742 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
17 | 14, 7, 16 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑅 unitVec 𝐼):𝐼⟶𝐵) |
18 | 17 | frnd 6731 | . . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵) |
19 | eqid 2725 | . . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
20 | frlmup.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑇) | |
21 | 2, 19, 20 | lmhmlsp 20946 | . . 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 582 | . 2 ⊢ (𝜑 → (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
23 | 2, 3 | lmhmf 20931 | . . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
25 | 24 | ffnd 6724 | . . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵) |
26 | fnima 6686 | . . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸) |
28 | eqid 2725 | . . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹) | |
29 | 1, 15, 28 | frlmlbs 21748 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
30 | 14, 7, 29 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ∈ (LBasis‘𝐹)) |
31 | 2, 28, 19 | lbssp 20976 | . . . . . 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 2731 | . . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) |
34 | 33 | imaeq2d 6064 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
35 | 27, 34 | eqtr3d 2767 | . 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
36 | imaco 6257 | . . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) | |
37 | 9 | ffnd 6724 | . . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼) |
38 | 17 | ffnd 6724 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼) |
39 | fnco 6673 | . . . . . . . 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 1368 | . . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼) |
41 | fvco2 6994 | . . . . . . . . 9 ⊢ (((𝑅 unitVec 𝐼) Fn 𝐼 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) | |
42 | 38, 41 | sylan 578 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢) = (𝐸‘((𝑅 unitVec 𝐼)‘𝑢))) |
43 | 6 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod) |
44 | 7 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋) |
45 | 8 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇)) |
46 | 9 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶) |
47 | simpr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼) | |
48 | 1, 2, 3, 4, 5, 43, 44, 45, 46, 47, 15 | frlmup2 21750 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢)) |
49 | 42, 48 | eqtr2d 2766 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢)) |
50 | 37, 40, 49 | eqfnfvd 7042 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼))) |
51 | 50 | imaeq1d 6063 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼)) |
52 | fnima 6686 | . . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴) | |
53 | 37, 52 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴) |
54 | 51, 53 | eqtr3d 2767 | . . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴) |
55 | fnima 6686 | . . . . . 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 6064 | . . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
58 | 36, 54, 57 | 3eqtr3a 2789 | . . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
59 | 58 | fveq2d 6900 | . 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
60 | 22, 35, 59 | 3eqtr4d 2775 | 1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ↦ cmpt 5232 ran crn 5679 “ cima 5681 ∘ ccom 5682 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 Basecbs 17183 Scalarcsca 17239 ·𝑠 cvsca 17240 Σg cgsu 17425 Ringcrg 20185 LModclmod 20755 LSpanclspn 20867 LMHom clmhm 20916 LBasisclbs 20971 freeLMod cfrlm 21697 unitVec cuvc 21733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-ghm 19176 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-nzr 20464 df-subrg 20520 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lmhm 20919 df-lbs 20972 df-sra 21070 df-rgmod 21071 df-dsmm 21683 df-frlm 21698 df-uvc 21734 |
This theorem is referenced by: ellspd 21753 indlcim 21791 lnrfg 42685 |
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