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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkr | Structured version Visualization version GIF version |
Description: The kernel of functional 𝐺 is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.) |
Ref | Expression |
---|---|
lshpkr.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpkr.a | ⊢ + = (+g‘𝑊) |
lshpkr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpkr.p | ⊢ ⊕ = (LSSum‘𝑊) |
lshpkr.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpkr.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpkr.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
lshpkr.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lshpkr.e | ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
lshpkr.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lshpkr.k | ⊢ 𝐾 = (Base‘𝐷) |
lshpkr.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lshpkr.g | ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) |
lshpkr.l | ⊢ 𝐿 = (LKer‘𝑊) |
Ref | Expression |
---|---|
lshpkr | ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpkr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | eqid 2731 | . . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
3 | lshpkr.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
4 | lshpkr.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 20666 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
7 | lshpkr.a | . . . . . 6 ⊢ + = (+g‘𝑊) | |
8 | lshpkr.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | lshpkr.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | lshpkr.h | . . . . . 6 ⊢ 𝐻 = (LSHyp‘𝑊) | |
11 | lshpkr.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
12 | lshpkr.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
13 | lshpkr.e | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | |
14 | lshpkr.d | . . . . . 6 ⊢ 𝐷 = (Scalar‘𝑊) | |
15 | lshpkr.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐷) | |
16 | lshpkr.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | lshpkr.g | . . . . . 6 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) | |
18 | 1, 7, 8, 9, 10, 4, 11, 12, 13, 14, 15, 16, 17, 2 | lshpkrcl 37789 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊)) |
19 | 1, 2, 3, 6, 18 | lkrssv 37769 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
20 | 19 | sseld 3977 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) → 𝑣 ∈ 𝑉)) |
21 | eqid 2731 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
22 | 21, 10, 6, 11 | lshplss 37654 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
23 | 1, 21 | lssel 20497 | . . . . 5 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
24 | 22, 23 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
25 | 24 | ex 413 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑈 → 𝑣 ∈ 𝑉)) |
26 | eqid 2731 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
27 | 1, 14, 26, 2, 3 | ellkr 37762 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ (LFnl‘𝑊)) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = (0g‘𝐷)))) |
28 | 4, 18, 27 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = (0g‘𝐷)))) |
29 | 28 | baibd 540 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
30 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
31 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
32 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
33 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
34 | 13 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
35 | 1, 7, 8, 9, 10, 30, 31, 32, 33, 34, 14, 15, 16, 26, 17 | lshpkrlem1 37783 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ 𝑈 ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
36 | 29, 35 | bitr4d 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
37 | 36 | ex 413 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑉 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈))) |
38 | 20, 25, 37 | pm5.21ndd 380 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
39 | 38 | eqrdv 2729 | 1 ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 {csn 4622 ↦ cmpt 5224 ‘cfv 6532 ℩crio 7348 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 Scalarcsca 17182 ·𝑠 cvsca 17183 0gc0g 17367 LSSumclsm 19466 LModclmod 20420 LSubSpclss 20491 LSpanclspn 20531 LVecclvec 20662 LSHypclsh 37648 LFnlclfn 37730 LKerclk 37758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 df-lshyp 37650 df-lfl 37731 df-lkr 37759 |
This theorem is referenced by: lshpkrex 37791 dochsnkr2 40147 |
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