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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkr0f | Structured version Visualization version GIF version |
Description: The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.) |
Ref | Expression |
---|---|
lkr0f.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkr0f.o | ⊢ 0 = (0g‘𝐷) |
lkr0f.v | ⊢ 𝑉 = (Base‘𝑊) |
lkr0f.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkr0f.k | ⊢ 𝐾 = (LKer‘𝑊) |
Ref | Expression |
---|---|
lkr0f | ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkr0f.d | . . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑊) | |
2 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
3 | lkr0f.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lkr0f.f | . . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | 1, 2, 3, 4 | lflf 33368 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
6 | ffn 5958 | . . . . . 6 ⊢ (𝐺:𝑉⟶(Base‘𝐷) → 𝐺 Fn 𝑉) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺 Fn 𝑉) |
8 | 7 | adantr 480 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → 𝐺 Fn 𝑉) |
9 | lkr0f.o | . . . . . . 7 ⊢ 0 = (0g‘𝐷) | |
10 | lkr0f.k | . . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊) | |
11 | 1, 9, 4, 10 | lkrval 33393 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
12 | 11 | eqeq1d 2612 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ (◡𝐺 “ { 0 }) = 𝑉)) |
13 | 12 | biimpa 500 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → (◡𝐺 “ { 0 }) = 𝑉) |
14 | fvex 6113 | . . . . . . 7 ⊢ (0g‘𝐷) ∈ V | |
15 | 9, 14 | eqeltri 2684 | . . . . . 6 ⊢ 0 ∈ V |
16 | 15 | fconst2 6375 | . . . . 5 ⊢ (𝐺:𝑉⟶{ 0 } ↔ 𝐺 = (𝑉 × { 0 })) |
17 | fconst4 6383 | . . . . 5 ⊢ (𝐺:𝑉⟶{ 0 } ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) | |
18 | 16, 17 | bitr3i 265 | . . . 4 ⊢ (𝐺 = (𝑉 × { 0 }) ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
19 | 8, 13, 18 | sylanbrc 695 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → 𝐺 = (𝑉 × { 0 })) |
20 | 19 | ex 449 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 → 𝐺 = (𝑉 × { 0 }))) |
21 | 18 | biimpi 205 | . . . . . 6 ⊢ (𝐺 = (𝑉 × { 0 }) → (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
22 | 21 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
23 | simpr 476 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 = (𝑉 × { 0 })) | |
24 | eqid 2610 | . . . . . . . . . . 11 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
25 | 1, 9, 3, 24 | lfl0f 33374 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
26 | 25 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
27 | 23, 26 | eqeltrd 2688 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 ∈ (LFnl‘𝑊)) |
28 | 1, 9, 24, 10 | lkrval 33393 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ (LFnl‘𝑊)) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
29 | 27, 28 | syldan 486 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
30 | 29 | eqeq1d 2612 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((𝐾‘𝐺) = 𝑉 ↔ (◡𝐺 “ { 0 }) = 𝑉)) |
31 | ffn 5958 | . . . . . . . . 9 ⊢ (𝐺:𝑉⟶{ 0 } → 𝐺 Fn 𝑉) | |
32 | 16, 31 | sylbir 224 | . . . . . . . 8 ⊢ (𝐺 = (𝑉 × { 0 }) → 𝐺 Fn 𝑉) |
33 | 32 | adantl 481 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 Fn 𝑉) |
34 | 33 | biantrurd 528 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((◡𝐺 “ { 0 }) = 𝑉 ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉))) |
35 | 30, 34 | bitrd 267 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((𝐾‘𝐺) = 𝑉 ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉))) |
36 | 22, 35 | mpbird 246 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐾‘𝐺) = 𝑉) |
37 | 36 | ex 449 | . . 3 ⊢ (𝑊 ∈ LMod → (𝐺 = (𝑉 × { 0 }) → (𝐾‘𝐺) = 𝑉)) |
38 | 37 | adantr 480 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺 = (𝑉 × { 0 }) → (𝐾‘𝐺) = 𝑉)) |
39 | 20, 38 | impbid 201 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ◡ccnv 5037 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 LModclmod 18686 LFnlclfn 33362 LKerclk 33390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ring 18372 df-lmod 18688 df-lfl 33363 df-lkr 33391 |
This theorem is referenced by: lkrscss 33403 eqlkr 33404 lkrshp 33410 lkrshp3 33411 lkrshpor 33412 lfl1dim 33426 lfl1dim2N 33427 lkr0f2 33466 lclkrlem1 35813 lclkrlem2j 35823 lclkr 35840 lclkrs 35846 mapd0 35972 |
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