![]() |
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 21836 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |
11 | eqid 2735 | . . . . . . . 8 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
12 | 11 | lmodring 20883 | . . . . . . 7 ⊢ (𝑇 ∈ LMod → (Scalar‘𝑇) ∈ Ring) |
13 | 6, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
14 | 8, 13 | eqeltrd 2839 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | eqid 2735 | . . . . . 6 ⊢ (𝑅 unitVec 𝐼) = (𝑅 unitVec 𝐼) | |
16 | 15, 1, 2 | uvcff 21829 | . . . . 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 6745 | . . 3 ⊢ (𝜑 → ran (𝑅 unitVec 𝐼) ⊆ 𝐵) |
19 | eqid 2735 | . . . 4 ⊢ (LSpan‘𝐹) = (LSpan‘𝐹) | |
20 | frlmup.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑇) | |
21 | 2, 19, 20 | lmhmlsp 21066 | . . 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 21051 | . . . . . 6 ⊢ (𝐸 ∈ (𝐹 LMHom 𝑇) → 𝐸:𝐵⟶𝐶) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
25 | 24 | ffnd 6738 | . . . 4 ⊢ (𝜑 → 𝐸 Fn 𝐵) |
26 | fnima 6699 | . . . 4 ⊢ (𝐸 Fn 𝐵 → (𝐸 “ 𝐵) = ran 𝐸) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = ran 𝐸) |
28 | eqid 2735 | . . . . . . . 8 ⊢ (LBasis‘𝐹) = (LBasis‘𝐹) | |
29 | 1, 15, 28 | frlmlbs 21835 | . . . . . . 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 21096 | . . . . . 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 2741 | . . . 4 ⊢ (𝜑 → 𝐵 = ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼))) |
34 | 33 | imaeq2d 6080 | . . 3 ⊢ (𝜑 → (𝐸 “ 𝐵) = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
35 | 27, 34 | eqtr3d 2777 | . 2 ⊢ (𝜑 → ran 𝐸 = (𝐸 “ ((LSpan‘𝐹)‘ran (𝑅 unitVec 𝐼)))) |
36 | imaco 6273 | . . . 4 ⊢ ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) | |
37 | 9 | ffnd 6738 | . . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝐼) |
38 | 17 | ffnd 6738 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 unitVec 𝐼) Fn 𝐼) |
39 | fnco 6687 | . . . . . . . 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 7006 | . . . . . . . . 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 21837 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐸‘((𝑅 unitVec 𝐼)‘𝑢)) = (𝐴‘𝑢)) |
49 | 42, 48 | eqtr2d 2776 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐼) → (𝐴‘𝑢) = ((𝐸 ∘ (𝑅 unitVec 𝐼))‘𝑢)) |
50 | 37, 40, 49 | eqfnfvd 7054 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐸 ∘ (𝑅 unitVec 𝐼))) |
51 | 50 | imaeq1d 6079 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼)) |
52 | fnima 6699 | . . . . . 6 ⊢ (𝐴 Fn 𝐼 → (𝐴 “ 𝐼) = ran 𝐴) | |
53 | 37, 52 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 “ 𝐼) = ran 𝐴) |
54 | 51, 53 | eqtr3d 2777 | . . . 4 ⊢ (𝜑 → ((𝐸 ∘ (𝑅 unitVec 𝐼)) “ 𝐼) = ran 𝐴) |
55 | fnima 6699 | . . . . . 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 6080 | . . . 4 ⊢ (𝜑 → (𝐸 “ ((𝑅 unitVec 𝐼) “ 𝐼)) = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
58 | 36, 54, 57 | 3eqtr3a 2799 | . . 3 ⊢ (𝜑 → ran 𝐴 = (𝐸 “ ran (𝑅 unitVec 𝐼))) |
59 | 58 | fveq2d 6911 | . 2 ⊢ (𝜑 → (𝐾‘ran 𝐴) = (𝐾‘(𝐸 “ ran (𝑅 unitVec 𝐼)))) |
60 | 22, 35, 59 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ↦ cmpt 5231 ran crn 5690 “ cima 5692 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 Σg cgsu 17487 Ringcrg 20251 LModclmod 20875 LSpanclspn 20987 LMHom clmhm 21036 LBasisclbs 21091 freeLMod cfrlm 21784 unitVec cuvc 21820 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-nzr 20530 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lbs 21092 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-uvc 21821 |
This theorem is referenced by: ellspd 21840 indlcim 21878 lnrfg 43108 |
Copyright terms: Public domain | W3C validator |