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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibval3N | Structured version Visualization version GIF version |
Description: Value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibval3.b | ⊢ 𝐵 = (Base‘𝐾) |
dibval3.l | ⊢ ≤ = (le‘𝐾) |
dibval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibval3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dibval3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dibval3.o | ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dibval3.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibval3N | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ({𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval3.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibval3.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dibval3.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dibval3.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | dibval3.o | . . 3 ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
6 | eqid 2821 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibval3.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 38274 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })) |
9 | dibval3.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | 1, 2, 3, 4, 9, 6 | diaval 38162 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((DIsoA‘𝐾)‘𝑊)‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
11 | 10 | xpeq1d 5579 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) = ({𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} × { 0 })) |
12 | 8, 11 | eqtrd 2856 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ({𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 {csn 4561 class class class wbr 5059 ↦ cmpt 5139 I cid 5454 × cxp 5548 ↾ cres 5552 ‘cfv 6350 Basecbs 16477 lecple 16566 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 DIsoAcdia 38158 DIsoBcdib 38268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-disoa 38159 df-dib 38269 |
This theorem is referenced by: (None) |
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