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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualkrsc | Structured version Visualization version GIF version |
Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.) |
Ref | Expression |
---|---|
ldualkrsc.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualkrsc.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualkrsc.o | ⊢ 0 = (0g‘𝑅) |
ldualkrsc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualkrsc.l | ⊢ 𝐿 = (LKer‘𝑊) |
ldualkrsc.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualkrsc.s | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualkrsc.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
ldualkrsc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualkrsc.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualkrsc.e | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
Ref | Expression |
---|---|
ldualkrsc | ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualkrsc.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | eqid 2738 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | ldualkrsc.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualkrsc.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
5 | eqid 2738 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | ldualkrsc.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
7 | ldualkrsc.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
8 | ldualkrsc.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
9 | ldualkrsc.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
10 | ldualkrsc.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ldualvs 37078 | . . 3 ⊢ (𝜑 → (𝑋 · 𝐺) = (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
12 | 11 | fveq2d 6760 | . 2 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘(𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})))) |
13 | ldualkrsc.l | . . 3 ⊢ 𝐿 = (LKer‘𝑊) | |
14 | ldualkrsc.o | . . 3 ⊢ 0 = (0g‘𝑅) | |
15 | ldualkrsc.e | . . 3 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
16 | 2, 3, 4, 5, 1, 13, 8, 10, 9, 14, 15 | lkrsc 37038 | . 2 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) = (𝐿‘𝐺)) |
17 | 12, 16 | eqtrd 2778 | 1 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 {csn 4558 × cxp 5578 ‘cfv 6418 (class class class)co 7255 ∘f cof 7509 Basecbs 16840 .rcmulr 16889 Scalarcsca 16891 ·𝑠 cvsca 16892 0gc0g 17067 LVecclvec 20279 LFnlclfn 36998 LKerclk 37026 LDualcld 37064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-drng 19908 df-lmod 20040 df-lvec 20280 df-lfl 36999 df-lkr 37027 df-ldual 37065 |
This theorem is referenced by: lclkrlem1 39447 lcfrlem31 39514 |
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