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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 2819 | . . . . . . 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 36191 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
6 | 5 | ffnd 6508 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺 Fn 𝑉) |
7 | 6 | adantr 483 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → 𝐺 Fn 𝑉) |
8 | lkr0f.o | . . . . . . 7 ⊢ 0 = (0g‘𝐷) | |
9 | lkr0f.k | . . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊) | |
10 | 1, 8, 4, 9 | lkrval 36216 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
11 | 10 | eqeq1d 2821 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ (◡𝐺 “ { 0 }) = 𝑉)) |
12 | 11 | biimpa 479 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → (◡𝐺 “ { 0 }) = 𝑉) |
13 | 8 | fvexi 6677 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | fconst2 6960 | . . . . 5 ⊢ (𝐺:𝑉⟶{ 0 } ↔ 𝐺 = (𝑉 × { 0 })) |
15 | fconst4 6969 | . . . . 5 ⊢ (𝐺:𝑉⟶{ 0 } ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) | |
16 | 14, 15 | bitr3i 279 | . . . 4 ⊢ (𝐺 = (𝑉 × { 0 }) ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
17 | 7, 12, 16 | sylanbrc 585 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ (𝐾‘𝐺) = 𝑉) → 𝐺 = (𝑉 × { 0 })) |
18 | 17 | ex 415 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 → 𝐺 = (𝑉 × { 0 }))) |
19 | 16 | biimpi 218 | . . . . . 6 ⊢ (𝐺 = (𝑉 × { 0 }) → (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
20 | 19 | adantl 484 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉)) |
21 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 = (𝑉 × { 0 })) | |
22 | eqid 2819 | . . . . . . . . . . 11 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
23 | 1, 8, 3, 22 | lfl0f 36197 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
24 | 23 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝑉 × { 0 }) ∈ (LFnl‘𝑊)) |
25 | 21, 24 | eqeltrd 2911 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 ∈ (LFnl‘𝑊)) |
26 | 1, 8, 22, 9 | lkrval 36216 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ (LFnl‘𝑊)) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
27 | 25, 26 | syldan 593 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
28 | 27 | eqeq1d 2821 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((𝐾‘𝐺) = 𝑉 ↔ (◡𝐺 “ { 0 }) = 𝑉)) |
29 | ffn 6507 | . . . . . . . . 9 ⊢ (𝐺:𝑉⟶{ 0 } → 𝐺 Fn 𝑉) | |
30 | 14, 29 | sylbir 237 | . . . . . . . 8 ⊢ (𝐺 = (𝑉 × { 0 }) → 𝐺 Fn 𝑉) |
31 | 30 | adantl 484 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 Fn 𝑉) |
32 | 31 | biantrurd 535 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((◡𝐺 “ { 0 }) = 𝑉 ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉))) |
33 | 28, 32 | bitrd 281 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((𝐾‘𝐺) = 𝑉 ↔ (𝐺 Fn 𝑉 ∧ (◡𝐺 “ { 0 }) = 𝑉))) |
34 | 20, 33 | mpbird 259 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → (𝐾‘𝐺) = 𝑉) |
35 | 34 | ex 415 | . . 3 ⊢ (𝑊 ∈ LMod → (𝐺 = (𝑉 × { 0 }) → (𝐾‘𝐺) = 𝑉)) |
36 | 35 | adantr 483 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺 = (𝑉 × { 0 }) → (𝐾‘𝐺) = 𝑉)) |
37 | 18, 36 | impbid 214 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐾‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 {csn 4559 × cxp 5546 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 Basecbs 16475 Scalarcsca 16560 0gc0g 16705 LModclmod 19626 LFnlclfn 36185 LKerclk 36213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19232 df-ring 19291 df-lmod 19628 df-lfl 36186 df-lkr 36214 |
This theorem is referenced by: lkrscss 36226 eqlkr 36227 lkrshp 36233 lkrshp3 36234 lkrshpor 36235 lfl1dim 36249 lfl1dim2N 36250 lkr0f2 36289 lclkrlem1 38634 lclkrlem2j 38644 lclkr 38661 lclkrs 38667 mapd0 38793 |
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