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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 21123 | . . . . . . 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 39078 | . . . . 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 2735 | . . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
13 | 1, 9, 2, 10, 11, 12, 6, 7 | lfl0sc 39064 | . . . . . . 7 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(0g‘𝐷)})) = (𝑉 × {(0g‘𝐷)})) |
14 | 13 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) = (𝐿‘(𝑉 × {(0g‘𝐷)}))) |
15 | eqid 2735 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝐷)}) = (𝑉 × {(0g‘𝐷)}) | |
16 | 9, 12, 1, 2 | lfl0f 39051 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (𝑉 × {(0g‘𝐷)}) ∈ 𝐹) |
17 | 9, 12, 1, 2, 3 | lkr0f 39076 | . . . . . . . 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 258 | . . . . . 6 ⊢ (𝜑 → (𝐿‘(𝑉 × {(0g‘𝐷)})) = 𝑉) |
20 | 14, 19 | eqtr2d 2776 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
21 | 8, 20 | sseqtrd 4036 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
22 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
23 | sneq 4641 | . . . . . . 7 ⊢ (𝑅 = (0g‘𝐷) → {𝑅} = {(0g‘𝐷)}) | |
24 | 23 | xpeq2d 5719 | . . . . . 6 ⊢ (𝑅 = (0g‘𝐷) → (𝑉 × {𝑅}) = (𝑉 × {(0g‘𝐷)})) |
25 | 24 | oveq2d 7447 | . . . . 5 ⊢ (𝑅 = (0g‘𝐷) → (𝐺 ∘f · (𝑉 × {𝑅})) = (𝐺 ∘f · (𝑉 × {(0g‘𝐷)}))) |
26 | 25 | fveq2d 6911 | . . . 4 ⊢ (𝑅 = (0g‘𝐷) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
27 | 26 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘(𝐺 ∘f · (𝑉 × {(0g‘𝐷)})))) |
28 | 22, 27 | sseqtrrd 4037 | . 2 ⊢ ((𝜑 ∧ 𝑅 = (0g‘𝐷)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
29 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑊 ∈ LVec) |
30 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝐺 ∈ 𝐹) |
31 | lkrsc.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
32 | 31 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ∈ 𝐾) |
33 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → 𝑅 ≠ (0g‘𝐷)) | |
34 | 1, 9, 10, 11, 2, 3, 29, 30, 32, 12, 33 | lkrsc 39079 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ≠ (0g‘𝐷)) → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
35 | eqimss2 4055 | . . 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 3027 | 1 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 {csn 4631 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 0gc0g 17486 LModclmod 20875 LVecclvec 21119 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-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-of 7697 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-grp 18967 df-minusg 18968 df-sbg 18969 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-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lvec 21120 df-lfl 39040 df-lkr 39068 |
This theorem is referenced by: lfl1dim 39103 lfl1dim2N 39104 lkrss 39150 |
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