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Mathbox for Norm Megill |
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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 2778 | . . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibval3.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 37731 | . 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 37619 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((DIsoA‘𝐾)‘𝑊)‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
11 | 10 | xpeq1d 5436 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) = ({𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} × { 0 })) |
12 | 8, 11 | eqtrd 2814 | 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 387 = wceq 1507 ∈ wcel 2050 {crab 3092 {csn 4441 class class class wbr 4929 ↦ cmpt 5008 I cid 5311 × cxp 5405 ↾ cres 5409 ‘cfv 6188 Basecbs 16339 lecple 16428 LHypclh 36571 LTrncltrn 36688 trLctrl 36745 DIsoAcdia 37615 DIsoBcdib 37725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-disoa 37616 df-dib 37726 |
This theorem is referenced by: (None) |
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