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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdval2 | Structured version Visualization version GIF version |
Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
lcdval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdval.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcdval.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdval.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcdval.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcdval.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcdval.k | ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) |
lcdval2.b | ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
Ref | Expression |
---|---|
lcdval2 | ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdval.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcdval.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcdval.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | lcdval.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | lcdval.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lcdval.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | lcdval.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcdval.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lcdval 41495 | . 2 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})) |
10 | lcdval2.b | . . 3 ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | 10 | oveq2i 7456 | . 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
12 | 9, 11 | eqtr4di 2792 | 1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 {crab 3438 ‘cfv 6572 (class class class)co 7445 ↾s cress 17282 LFnlclfn 38962 LKerclk 38990 LDualcld 39028 LHypclh 39890 DVecHcdvh 40984 ocHcoch 41253 LCDualclcd 41492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-lcdual 41493 |
This theorem is referenced by: lcdvbase 41499 lcdlss 41525 |
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