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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 2821 | . . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
3 | lshpkr.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
4 | lshpkr.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 19878 | . . . . . 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 36267 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊)) |
19 | 1, 2, 3, 6, 18 | lkrssv 36247 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
20 | 19 | sseld 3966 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) → 𝑣 ∈ 𝑉)) |
21 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
22 | 21, 10, 6, 11 | lshplss 36132 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
23 | 1, 21 | lssel 19709 | . . . . 5 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
24 | 22, 23 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
25 | 24 | ex 415 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑈 → 𝑣 ∈ 𝑉)) |
26 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
27 | 1, 14, 26, 2, 3 | ellkr 36240 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ (LFnl‘𝑊)) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = (0g‘𝐷)))) |
28 | 4, 18, 27 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = (0g‘𝐷)))) |
29 | 28 | baibd 542 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
30 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
31 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
32 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
33 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
34 | 13 | adantr 483 | . . . . . 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 36261 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ 𝑈 ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
36 | 29, 35 | bitr4d 284 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
37 | 36 | ex 415 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑉 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈))) |
38 | 20, 25, 37 | pm5.21ndd 383 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
39 | 38 | eqrdv 2819 | 1 ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {csn 4567 ↦ cmpt 5146 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LSSumclsm 18759 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874 LSHypclsh 36126 LFnlclfn 36208 LKerclk 36236 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lshyp 36128 df-lfl 36209 df-lkr 36237 |
This theorem is referenced by: lshpkrex 36269 dochsnkr2 38624 |
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