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Mirrors > Home > MPE Home > Th. List > Mathboxes > diaval | Structured version Visualization version GIF version |
Description: The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-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 |
---|---|
diaval | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diaval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | diaval.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | diaval.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | diaval.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | diaval.r | . . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
6 | diaval.i | . . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | diafval 36822 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})) |
8 | 7 | adantr 472 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})) |
9 | 8 | fveq1d 6354 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋)) |
10 | simpr 479 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) | |
11 | breq1 4807 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊)) | |
12 | 11 | elrab 3504 | . . . 4 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) |
13 | 10, 12 | sylibr 224 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊}) |
14 | breq2 4808 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑋)) | |
15 | 14 | rabbidv 3329 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
16 | eqid 2760 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) | |
17 | fvex 6362 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
18 | 4, 17 | eqeltri 2835 | . . . . 5 ⊢ 𝑇 ∈ V |
19 | 18 | rabex 4964 | . . . 4 ⊢ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} ∈ V |
20 | 15, 16, 19 | fvmpt 6444 | . . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} → ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
21 | 13, 20 | syl 17 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ((𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
22 | 9, 21 | eqtrd 2794 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 {crab 3054 Vcvv 3340 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 Basecbs 16059 lecple 16150 LHypclh 35773 LTrncltrn 35890 trLctrl 35948 DIsoAcdia 36819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-disoa 36820 |
This theorem is referenced by: diaelval 36824 diass 36833 diaord 36838 dia0 36843 dia1N 36844 diassdvaN 36851 dia1dim 36852 cdlemm10N 36909 dibval3N 36937 dihwN 37080 |
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