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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 18927 | . . . . . . 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 33401 | . . . . 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 2610 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 1, 9, 2, 10, 11, 12, 6, 7 | lfl0sc 33387 | . . . . . . 7 ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)})) |
14 | 13 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)}))) = (𝐿‘(𝑉 × {(0g‘𝐷)}))) |
15 | eqid 2610 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}) | |
16 | 9, 12, 1, 2 | lfl0f 33374 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) |
17 | 6, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) |
18 | 9, 12, 1, 2, 3 | lkr0f 33399 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
19 | 6, 17, 18 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → ((𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉 ↔ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}))) |
20 | 15, 19 | mpbiri 247 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉) |
21 | 14, 20 | eqtr2d 2645 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
22 | 8, 21 | sseqtrd 3604 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
23 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
24 | sneq 4135 | . . . . . . 7 ⊢ (𝑅 = (0g‘𝐷) → {𝑅} = {(0g‘𝐷)}) | |
25 | 24 | xpeq2d 5063 | . . . . . 6 ⊢ (𝑅 = (0g‘𝐷) → (𝑉 × {𝑅}) = (𝑉 × {(0g‘𝐷)})) |
26 | 25 | oveq2d 6565 | . . . . 5 ⊢ (𝑅 = (0g‘𝐷) → (𝐺 ∘𝑓 · (𝑉 × {𝑅})) = (𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)}))) |
27 | 26 | fveq2d 6107 | . . . 4 ⊢ (𝑅 = (0g‘𝐷) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
28 | 27 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {(0g‘𝐷)})))) |
29 | 23, 28 | sseqtr4d 3605 | . 2 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) |
30 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑊 ∈ LVec) |
31 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝐺 ∈ 𝐹) |
32 | lkrsc.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
33 | 32 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ∈ 𝐾) |
34 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ≠ (0g‘𝐷)) | |
35 | 1, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34 | lkrsc 33402 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
36 | eqimss2 3621 | . . 3 ⊢ ((𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) |
38 | 29, 37 | pm2.61dane 2869 | 1 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 .rcmulr 15769 Scalarcsca 15771 0gc0g 15923 LModclmod 18686 LVecclvec 18923 LFnlclfn 33362 LKerclk 33390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lvec 18924 df-lfl 33363 df-lkr 33391 |
This theorem is referenced by: lfl1dim 33426 lfl1dim2N 33427 lkrss 33473 |
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