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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdvalc | Structured version Visualization version GIF version |
Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.) |
Ref | Expression |
---|---|
mapdval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdval.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
mapdval.f | ⊢ 𝐹 = (LFnl‘𝑈) |
mapdval.l | ⊢ 𝐿 = (LKer‘𝑈) |
mapdval.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
mapdval.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdval.k | ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) |
mapdval.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
mapdvalc.c | ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} |
Ref | Expression |
---|---|
mapdvalc | ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdval.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdval.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑈) | |
4 | mapdval.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
5 | mapdval.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
6 | mapdval.o | . . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
7 | mapdval.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
8 | mapdval.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻)) | |
9 | mapdval.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mapdval 35935 | . 2 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)}) |
11 | anass 679 | . . . 4 ⊢ (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))) | |
12 | mapdvalc.c | . . . . . . . 8 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} | |
13 | 12 | lcfl1lem 35798 | . . . . . . 7 ⊢ (𝑓 ∈ 𝐶 ↔ (𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓))) |
14 | 13 | anbi1i 727 | . . . . . 6 ⊢ ((𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ ((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) |
15 | 14 | bicomi 213 | . . . . 5 ⊢ (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))) |
17 | 11, 16 | syl5bbr 273 | . . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))) |
18 | 17 | rabbidva2 3162 | . 2 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇}) |
19 | 10, 18 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 ‘cfv 5804 LSubSpclss 18753 LFnlclfn 33362 LKerclk 33390 LHypclh 34288 DVecHcdvh 35385 ocHcoch 35654 mapdcmpd 35931 |
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-mapd 35932 |
This theorem is referenced by: mapdval2N 35937 mapdordlem2 35944 mapdrval 35954 |
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