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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaelval | Structured version Visualization version GIF version |
Description: Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.) |
Ref | Expression |
---|---|
diaval.b | ⊢ 𝐵 = (Base‘𝐾) |
diaval.l | ⊢ ≤ = (le‘𝐾) |
diaval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
diaval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
diaval.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
diaval.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
diaelval | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diaval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | diaval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | diaval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diaval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | diaval.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
6 | diaval.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | diaval 38167 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
8 | 7 | eleq2d 2898 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})) |
9 | fveq2 6669 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹)) | |
10 | 9 | breq1d 5075 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑅‘𝑓) ≤ 𝑋 ↔ (𝑅‘𝐹) ≤ 𝑋)) |
11 | 10 | elrab 3679 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)) |
12 | 8, 11 | syl6bb 289 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5065 ‘cfv 6354 Basecbs 16482 lecple 16571 LHypclh 37119 LTrncltrn 37236 trLctrl 37293 DIsoAcdia 38163 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-disoa 38164 |
This theorem is referenced by: dian0 38174 diatrl 38179 dialss 38181 diaglbN 38190 dibelval3 38282 dibopelval3 38283 diblss 38305 |
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