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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 2823 | . . . . . . 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 36201 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝐷)) |
6 | 5 | ffnd 6517 | . . . . 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 36226 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 })) |
11 | 10 | eqeq1d 2825 | . . . . 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 6686 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | fconst2 6969 | . . . . 5 ⊢ (𝐺:𝑉⟶{ 0 } ↔ 𝐺 = (𝑉 × { 0 })) |
15 | fconst4 6979 | . . . . 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 2823 | . . . . . . . . . . 11 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
23 | 1, 8, 3, 22 | lfl0f 36207 | . . . . . . . . . 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 2915 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → 𝐺 ∈ (LFnl‘𝑊)) |
26 | 1, 8, 22, 9 | lkrval 36226 | . . . . . . . 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 2825 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 = (𝑉 × { 0 })) → ((𝐾‘𝐺) = 𝑉 ↔ (◡𝐺 “ { 0 }) = 𝑉)) |
29 | ffn 6516 | . . . . . . . . 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 1537 ∈ wcel 2114 {csn 4569 × cxp 5555 ◡ccnv 5556 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 LModclmod 19636 LFnlclfn 36195 LKerclk 36223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mgp 19242 df-ring 19301 df-lmod 19638 df-lfl 36196 df-lkr 36224 |
This theorem is referenced by: lkrscss 36236 eqlkr 36237 lkrshp 36243 lkrshp3 36244 lkrshpor 36245 lfl1dim 36259 lfl1dim2N 36260 lkr0f2 36299 lclkrlem1 38644 lclkrlem2j 38654 lclkr 38671 lclkrs 38677 mapd0 38803 |
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