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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 39340 | . 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 7224 | . 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
12 | 9, 11 | eqtr4di 2796 | 1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3065 ‘cfv 6380 (class class class)co 7213 ↾s cress 16784 LFnlclfn 36808 LKerclk 36836 LDualcld 36874 LHypclh 37735 DVecHcdvh 38829 ocHcoch 39098 LCDualclcd 39337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-lcdual 39338 |
This theorem is referenced by: lcdvbase 39344 lcdlss 39370 |
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