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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st | Structured version Visualization version GIF version |
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) |
Ref | Expression |
---|---|
dibelval1.b | ⊢ 𝐵 = (Base‘𝐾) |
dibelval1.l | ⊢ ≤ = (le‘𝐾) |
dibelval1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dibelval1.j | ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) |
dibelval1.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dibelval1st | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (𝐽‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dibelval1.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | dibelval1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2736 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
5 | eqid 2736 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) | |
6 | dibelval1.j | . . . . 5 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibelval1.i | . . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 41124 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) |
9 | 8 | eleq2d 2826 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}))) |
10 | 9 | biimp3a 1471 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))})) |
11 | xp1st 8042 | . 2 ⊢ (𝑌 ∈ ((𝐽‘𝑋) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))}) → (1st ‘𝑌) ∈ (𝐽‘𝑋)) | |
12 | 10, 11 | syl 17 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (𝐽‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {csn 4624 class class class wbr 5141 ↦ cmpt 5223 I cid 5575 × cxp 5681 ↾ cres 5685 ‘cfv 6559 1st c1st 8008 Basecbs 17243 lecple 17300 LHypclh 39964 LTrncltrn 40081 DIsoAcdia 41008 DIsoBcdib 41118 |
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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-1st 8010 df-disoa 41009 df-dib 41119 |
This theorem is referenced by: dibelval1st1 41130 dibelval1st2N 41131 diblss 41150 |
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