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Mathbox for Norm Megill |
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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 | β’ π = (Baseβπ) |
lkr0f2.f | β’ πΉ = (LFnlβπ) |
lkr0f2.k | β’ πΎ = (LKerβπ) |
lkr0f2.d | β’ π· = (LDualβπ) |
lkr0f2.o | β’ 0 = (0gβπ·) |
lkr0f2.w | β’ (π β π β LMod) |
lkr0f2.g | β’ (π β πΊ β πΉ) |
Ref | Expression |
---|---|
lkr0f2 | β’ (π β ((πΎβπΊ) = π β πΊ = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkr0f2.w | . . 3 β’ (π β π β LMod) | |
2 | lkr0f2.g | . . 3 β’ (π β πΊ β πΉ) | |
3 | eqid 2724 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2724 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
5 | lkr0f2.v | . . . 4 β’ π = (Baseβπ) | |
6 | lkr0f2.f | . . . 4 β’ πΉ = (LFnlβπ) | |
7 | lkr0f2.k | . . . 4 β’ πΎ = (LKerβπ) | |
8 | 3, 4, 5, 6, 7 | lkr0f 38468 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β πΊ = (π Γ {(0gβ(Scalarβπ))}))) |
9 | 1, 2, 8 | syl2anc 583 | . 2 β’ (π β ((πΎβπΊ) = π β πΊ = (π Γ {(0gβ(Scalarβπ))}))) |
10 | lkr0f2.d | . . . 4 β’ π· = (LDualβπ) | |
11 | lkr0f2.o | . . . 4 β’ 0 = (0gβπ·) | |
12 | 5, 3, 4, 10, 11, 1 | ldual0v 38524 | . . 3 β’ (π β 0 = (π Γ {(0gβ(Scalarβπ))})) |
13 | 12 | eqeq2d 2735 | . 2 β’ (π β (πΊ = 0 β πΊ = (π Γ {(0gβ(Scalarβπ))}))) |
14 | 9, 13 | bitr4d 282 | 1 β’ (π β ((πΎβπΊ) = π β πΊ = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 {csn 4621 Γ cxp 5665 βcfv 6534 Basecbs 17149 Scalarcsca 17205 0gc0g 17390 LModclmod 20702 LFnlclfn 38431 LKerclk 38459 LDualcld 38497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sca 17218 df-vsca 17219 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-lmod 20704 df-lfl 38432 df-lkr 38460 df-ldual 38498 |
This theorem is referenced by: lkrpssN 38537 lcfl8b 40879 lcfrlem9 40925 |
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