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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 35896 | . 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 6560 | . 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
12 | 9, 11 | syl6eqr 2662 | 1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ‘cfv 5804 (class class class)co 6549 ↾s cress 15696 LFnlclfn 33362 LKerclk 33390 LDualcld 33428 LHypclh 34288 DVecHcdvh 35385 ocHcoch 35654 LCDualclcd 35893 |
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-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-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-reu 2903 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-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-lcdual 35894 |
This theorem is referenced by: lcdvbase 35900 lcdlss 35926 |
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