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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 19556 | . . . . . . 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 35713 | . . . . 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 2793 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 1, 9, 2, 10, 11, 12, 6, 7 | lfl0sc 35699 | . . . . . . 7 ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)})) |
14 | 13 | fveq2d 6534 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)}))) = (𝐿‘(𝑉 × {(0g‘𝐷)}))) |
15 | eqid 2793 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}) | |
16 | 9, 12, 1, 2 | lfl0f 35686 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) |
17 | 9, 12, 1, 2, 3 | lkr0f 35711 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
18 | 6, 16, 17 | syl2anc2 585 | . . . . . . 7 ⊢ (𝜑 → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
19 | 15, 18 | mpbiri 259 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉) |
20 | 14, 19 | eqtr2d 2830 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
21 | 8, 20 | sseqtrd 3923 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
22 | 21 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
23 | sneq 4476 | . . . . . . 7 ⊢ (𝑅 = (0g‘𝐷) → {𝑅} = {(0g‘𝐷)}) | |
24 | 23 | xpeq2d 5465 | . . . . . 6 ⊢ (𝑅 = (0g‘𝐷) → (𝑉 × {𝑅}) = (𝑉 × {(0g‘𝐷)})) |
25 | 24 | oveq2d 7023 | . . . . 5 ⊢ (𝑅 = (0g‘𝐷) → (𝐺 ∘𝑓 · (𝑉 × {𝑅})) = (𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)}))) |
26 | 25 | fveq2d 6534 | . . . 4 ⊢ (𝑅 = (0g‘𝐷) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
27 | 26 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
28 | 22, 27 | sseqtr4d 3924 | . 2 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) |
29 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑊 ∈ LVec) |
30 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝐺 ∈ 𝐹) |
31 | lkrsc.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
32 | 31 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ∈ 𝐾) |
33 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ≠ (0g‘𝐷)) | |
34 | 1, 9, 10, 11, 2, 3, 29, 30, 32, 12, 33 | lkrsc 35714 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
35 | eqimss2 3940 | . . 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 3070 | 1 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ⊆ wss 3854 {csn 4466 × cxp 5433 ‘cfv 6217 (class class class)co 7007 ∘𝑓 cof 7256 Basecbs 16300 .rcmulr 16383 Scalarcsca 16385 0gc0g 16530 LModclmod 19312 LVecclvec 19552 LFnlclfn 35674 LKerclk 35702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-0g 16532 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-grp 17852 df-minusg 17853 df-sbg 17854 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-drng 19182 df-lmod 19314 df-lss 19382 df-lvec 19553 df-lfl 35675 df-lkr 35703 |
This theorem is referenced by: lfl1dim 35738 lfl1dim2N 35739 lkrss 35785 |
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