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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 38605 | . 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 7156 | . 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
12 | 9, 11 | syl6eqr 2871 | 1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ‘cfv 6348 (class class class)co 7145 ↾s cress 16472 LFnlclfn 36073 LKerclk 36101 LDualcld 36139 LHypclh 37000 DVecHcdvh 38094 ocHcoch 38363 LCDualclcd 38602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-lcdual 38603 |
This theorem is referenced by: lcdvbase 38609 lcdlss 38635 |
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