![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2735 | . . . . 5 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
3 | lshpkr.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
4 | lshpkr.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 21123 | . . . . . 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 39098 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑊)) |
19 | 1, 2, 3, 6, 18 | lkrssv 39078 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
20 | 19 | sseld 3994 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) → 𝑣 ∈ 𝑉)) |
21 | eqid 2735 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
22 | 21, 10, 6, 11 | lshplss 38963 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
23 | 1, 21 | lssel 20953 | . . . . 5 ⊢ ((𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
24 | 22, 23 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑉) |
25 | 24 | ex 412 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑈 → 𝑣 ∈ 𝑉)) |
26 | eqid 2735 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
27 | 1, 14, 26, 2, 3 | ellkr 39071 | . . . . . . 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 539 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
30 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
31 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ 𝐻) |
32 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑍 ∈ 𝑉) |
33 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
34 | 13 | adantr 480 | . . . . . 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 39092 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ 𝑈 ↔ (𝐺‘𝑣) = (0g‘𝐷))) |
36 | 29, 35 | bitr4d 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
37 | 36 | ex 412 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝑉 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈))) |
38 | 20, 25, 37 | pm5.21ndd 379 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ 𝑣 ∈ 𝑈)) |
39 | 38 | eqrdv 2733 | 1 ⊢ (𝜑 → (𝐿‘𝐺) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LSSumclsm 19667 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 LVecclvec 21119 LSHypclsh 38957 LFnlclfn 39039 LKerclk 39067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lshyp 38959 df-lfl 39040 df-lkr 39068 |
This theorem is referenced by: lshpkrex 39100 dochsnkr2 41456 |
Copyright terms: Public domain | W3C validator |