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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 41034 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) | 
| 8 | 7 | eleq2d 2827 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ 𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋})) | 
| 9 | fveq2 6906 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹)) | |
| 10 | 9 | breq1d 5153 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑅‘𝑓) ≤ 𝑋 ↔ (𝑅‘𝐹) ≤ 𝑋)) | 
| 11 | 10 | elrab 3692 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋)) | 
| 12 | 8, 11 | bitrdi 287 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 DIsoAcdia 41030 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-disoa 41031 | 
| This theorem is referenced by: dian0 41041 diatrl 41046 dialss 41048 diaglbN 41057 dibelval3 41149 dibopelval3 41150 diblss 41172 | 
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