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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 21819 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) | 
| 11 | eqid 2736 | . . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
| 12 | 11 | lmodring 20867 | . . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) | 
| 13 | 6, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) | 
| 14 | 8, 13 | eqeltrd 2840 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 15 | eqid 2736 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
| 16 | 15, 1, 2 | uvcff 21812 | . . . . 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 6743 | . . 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 21049 | . . 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 21034 | . . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶) | 
| 24 | 10, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶) | 
| 25 | 24 | ffnd 6736 | . . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵) | 
| 26 | fnima 6697 | . . . 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 21818 | . . . . . . 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 21079 | . . . . . 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 6077 | . . 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 6270 | . . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) | |
| 37 | 9 | ffnd 6736 | . . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼) | 
| 38 | 17 | ffnd 6736 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼) | 
| 39 | fnco 6685 | . . . . . . . 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 1372 | . . . . . . 7 ⊢ (𝜑 → (𝐸 ∘ (𝑅 unitVec 𝐼)) Fn 𝐼) | 
| 41 | fvco2 7005 | . . . . . . . . 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 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑇 ∈ LMod) | 
| 44 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐼 ∈ 𝑋) | 
| 45 | 8 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑅 = (Scalar‘𝑇)) | 
| 46 | 9 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝐴:𝐼⟶𝐶) | 
| 47 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → 𝑢 ∈ 𝐼) | |
| 48 | 1, 2, 3, 4, 5, 43, 44, 45, 46, 47, 15 | frlmup2 21820 | . . . . . . . 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 7053 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼))) | 
| 51 | 50 | imaeq1d 6076 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼)) | 
| 52 | fnima 6697 | . . . . . 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 6697 | . . . . . 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 6077 | . . . 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 6909 | . 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 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ↦ cmpt 5224 ran crn 5685 “ cima 5687 ∘ ccom 5688 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 Σg cgsu 17486 Ringcrg 20231 LModclmod 20859 LSpanclspn 20970 LMHom clmhm 21019 LBasisclbs 21074 freeLMod cfrlm 21767 unitVec cuvc 21803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-nzr 20514 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lmhm 21022 df-lbs 21075 df-sra 21173 df-rgmod 21174 df-dsmm 21753 df-frlm 21768 df-uvc 21804 | 
| This theorem is referenced by: ellspd 21823 indlcim 21861 lnrfg 43136 | 
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