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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkr0f2 | Structured version Visualization version GIF version |
Description: The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
lkr0f2.v | ⊢ 𝑉 = (Base‘𝑊) |
lkr0f2.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkr0f2.k | ⊢ 𝐾 = (LKer‘𝑊) |
lkr0f2.d | ⊢ 𝐷 = (LDual‘𝑊) |
lkr0f2.o | ⊢ 0 = (0g‘𝐷) |
lkr0f2.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lkr0f2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lkr0f2 | ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkr0f2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lkr0f2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
3 | eqid 2778 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2778 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | lkr0f2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lkr0f2.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
7 | lkr0f2.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
8 | 3, 4, 5, 6, 7 | lkr0f 35248 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
9 | 1, 2, 8 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
10 | lkr0f2.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
11 | lkr0f2.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
12 | 5, 3, 4, 10, 11, 1 | ldual0v 35304 | . . 3 ⊢ (𝜑 → 0 = (𝑉 × {(0g‘(Scalar‘𝑊))})) |
13 | 12 | eqeq2d 2788 | . 2 ⊢ (𝜑 → (𝐺 = 0 ↔ 𝐺 = (𝑉 × {(0g‘(Scalar‘𝑊))}))) |
14 | 9, 13 | bitr4d 274 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 {csn 4398 × cxp 5353 ‘cfv 6135 Basecbs 16255 Scalarcsca 16341 0gc0g 16486 LModclmod 19255 LFnlclfn 35211 LKerclk 35239 LDualcld 35277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-sca 16354 df-vsca 16355 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lfl 35212 df-lkr 35240 df-ldual 35278 |
This theorem is referenced by: lkrpssN 35317 lcfl8b 37658 lcfrlem9 37704 |
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