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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrscss | Structured version Visualization version GIF version |
Description: The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.) |
Ref | Expression |
---|---|
lkrsc.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrsc.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrsc.k | ⊢ 𝐾 = (Base‘𝐷) |
lkrsc.t | ⊢ · = (.r‘𝐷) |
lkrsc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrsc.l | ⊢ 𝐿 = (LKer‘𝑊) |
lkrsc.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrsc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lkrsc.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
Ref | Expression |
---|---|
lkrscss | ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrsc.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lkrsc.f | . . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊) | |
3 | lkrsc.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑊) | |
4 | lkrsc.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 21003 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
7 | lkrsc.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
8 | 1, 2, 3, 6, 7 | lkrssv 38698 | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
9 | lkrsc.d | . . . . . . . 8 ⊢ 𝐷 = (Scalar‘𝑊) | |
10 | lkrsc.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐷) | |
11 | lkrsc.t | . . . . . . . 8 ⊢ · = (.r‘𝐷) | |
12 | eqid 2725 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 1, 9, 2, 10, 11, 12, 6, 7 | lfl0sc 38684 | . . . . . . 7 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)})) |
14 | 13 | fveq2d 6900 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) = (𝐿‘(𝑉 × {(0g‘𝐷)}))) |
15 | eqid 2725 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}) | |
16 | 9, 12, 1, 2 | lfl0f 38671 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) |
17 | 9, 12, 1, 2, 3 | lkr0f 38696 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
18 | 6, 16, 17 | syl2anc2 583 | . . . . . . 7 ⊢ (𝜑 → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
19 | 15, 18 | mpbiri 257 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉) |
20 | 14, 19 | eqtr2d 2766 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
21 | 8, 20 | sseqtrd 4017 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
22 | 21 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
23 | sneq 4640 | . . . . . . 7 ⊢ (𝑅 = (0g‘𝐷) → {𝑅} = {(0g‘𝐷)}) | |
24 | 23 | xpeq2d 5708 | . . . . . 6 ⊢ (𝑅 = (0g‘𝐷) → (𝑉 × {𝑅}) = (𝑉 × {(0g‘𝐷)})) |
25 | 24 | oveq2d 7435 | . . . . 5 ⊢ (𝑅 = (0g‘𝐷) → (𝐺 ∘f · (𝑉 × {𝑅})) = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) |
26 | 25 | fveq2d 6900 | . . . 4 ⊢ (𝑅 = (0g‘𝐷) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
27 | 26 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
28 | 22, 27 | sseqtrrd 4018 | . 2 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
29 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑊 ∈ LVec) |
30 | 7 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝐺 ∈ 𝐹) |
31 | lkrsc.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
32 | 31 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ∈ 𝐾) |
33 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ≠ (0g‘𝐷)) | |
34 | 1, 9, 10, 11, 2, 3, 29, 30, 32, 12, 33 | lkrsc 38699 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
35 | eqimss2 4036 | . . 3 ⊢ ((𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) | |
36 | 34, 35 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
37 | 28, 36 | pm2.61dane 3018 | 1 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 {csn 4630 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 Basecbs 17183 .rcmulr 17237 Scalarcsca 17239 0gc0g 17424 LModclmod 20755 LVecclvec 20999 LFnlclfn 38659 LKerclk 38687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-drng 20638 df-lmod 20757 df-lss 20828 df-lvec 21000 df-lfl 38660 df-lkr 38688 |
This theorem is referenced by: lfl1dim 38723 lfl1dim2N 38724 lkrss 38770 |
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