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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 2824 | . . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
7 | dibelval2nd.i | . . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 38284 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })) |
9 | 8 | eleq2d 2901 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))) |
10 | 9 | biimp3a 1465 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })) |
11 | xp2nd 7725 | . 2 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd ‘𝑌) ∈ { 0 }) | |
12 | elsni 4587 | . 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 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {csn 4570 class class class wbr 5069 ↦ cmpt 5149 I cid 5462 × cxp 5556 ↾ cres 5560 ‘cfv 6358 2nd c2nd 7691 Basecbs 16486 lecple 16575 LHypclh 37124 LTrncltrn 37241 DIsoAcdia 38168 DIsoBcdib 38278 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-2nd 7693 df-disoa 38169 df-dib 38279 |
This theorem is referenced by: diblss 38310 |
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