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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval2nd | 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 | 
|---|---|
| dibelval2nd.b | ⊢ 𝐵 = (Base‘𝐾) | 
| dibelval2nd.l | ⊢ ≤ = (le‘𝐾) | 
| dibelval2nd.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| dibelval2nd.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | 
| dibelval2nd.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | 
| dibelval2nd.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | 
| Ref | Expression | 
|---|---|
| dibelval2nd | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (2nd ‘𝑌) = 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dibelval2nd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | dibelval2nd.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
| 3 | dibelval2nd.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 4 | dibelval2nd.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 5 | dibelval2nd.o | . . . . 5 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
| 6 | eqid 2736 | . . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
| 7 | dibelval2nd.i | . . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 41147 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })) | 
| 9 | 8 | eleq2d 2826 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))) | 
| 10 | 9 | biimp3a 1470 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })) | 
| 11 | xp2nd 8048 | . 2 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd ‘𝑌) ∈ { 0 }) | |
| 12 | elsni 4642 | . 2 ⊢ ((2nd ‘𝑌) ∈ { 0 } → (2nd ‘𝑌) = 0 ) | |
| 13 | 10, 11, 12 | 3syl 18 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (2nd ‘𝑌) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 I cid 5576 × cxp 5682 ↾ cres 5686 ‘cfv 6560 2nd c2nd 8014 Basecbs 17248 lecple 17305 LHypclh 39987 LTrncltrn 40104 DIsoAcdia 41031 DIsoBcdib 41141 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-2nd 8016 df-disoa 41032 df-dib 41142 | 
| This theorem is referenced by: diblss 41173 | 
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