![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqlkr2 | Structured version Visualization version GIF version |
Description: Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.) |
Ref | Expression |
---|---|
eqlkr.d | ⊢ 𝐷 = (Scalar‘𝑊) |
eqlkr.k | ⊢ 𝐾 = (Base‘𝐷) |
eqlkr.t | ⊢ · = (.r‘𝐷) |
eqlkr.v | ⊢ 𝑉 = (Base‘𝑊) |
eqlkr.f | ⊢ 𝐹 = (LFnl‘𝑊) |
eqlkr.l | ⊢ 𝐿 = (LKer‘𝑊) |
Ref | Expression |
---|---|
eqlkr2 | ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘f · (𝑉 × {𝑟}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqlkr.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑊) | |
2 | eqlkr.k | . . 3 ⊢ 𝐾 = (Base‘𝐷) | |
3 | eqlkr.t | . . 3 ⊢ · = (.r‘𝐷) | |
4 | eqlkr.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqlkr.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
6 | eqlkr.l | . . 3 ⊢ 𝐿 = (LKer‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | eqlkr 36395 | . 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)) |
8 | 4 | fvexi 6659 | . . . . 5 ⊢ 𝑉 ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝑉 ∈ V) |
10 | simpl1 1188 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝑊 ∈ LVec) | |
11 | simpl2l 1223 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺 ∈ 𝐹) | |
12 | 1, 2, 4, 5 | lflf 36359 | . . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
13 | 10, 11, 12 | syl2anc 587 | . . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺:𝑉⟶𝐾) |
14 | 13 | ffnd 6488 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐺 Fn 𝑉) |
15 | vex 3444 | . . . . 5 ⊢ 𝑟 ∈ V | |
16 | fnconstg 6541 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑉 × {𝑟}) Fn 𝑉) | |
17 | 15, 16 | mp1i 13 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → (𝑉 × {𝑟}) Fn 𝑉) |
18 | simpl2r 1224 | . . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻 ∈ 𝐹) | |
19 | 1, 2, 4, 5 | lflf 36359 | . . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶𝐾) |
20 | 10, 18, 19 | syl2anc 587 | . . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻:𝑉⟶𝐾) |
21 | 20 | ffnd 6488 | . . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → 𝐻 Fn 𝑉) |
22 | eqidd 2799 | . . . 4 ⊢ ((((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
23 | 15 | fvconst2 6943 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 → ((𝑉 × {𝑟})‘𝑥) = 𝑟) |
24 | 23 | adantl 485 | . . . 4 ⊢ ((((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → ((𝑉 × {𝑟})‘𝑥) = 𝑟) |
25 | 9, 14, 17, 21, 22, 24 | offveqb 7411 | . . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝑟 ∈ 𝐾) → (𝐻 = (𝐺 ∘f · (𝑉 × {𝑟})) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))) |
26 | 25 | rexbidva 3255 | . 2 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → (∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘f · (𝑉 × {𝑟})) ↔ ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))) |
27 | 7, 26 | mpbird 260 | 1 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 𝐻 = (𝐺 ∘f · (𝑉 × {𝑟}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 {csn 4525 × cxp 5517 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 .rcmulr 16558 Scalarcsca 16560 LVecclvec 19867 LFnlclfn 36353 LKerclk 36381 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lvec 19868 df-lfl 36354 df-lkr 36382 |
This theorem is referenced by: lfl1dim 36417 lfl1dim2N 36418 eqlkr4 36461 |
Copyright terms: Public domain | W3C validator |