Theorem dibelval2nd 41926

Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)

Hypotheses

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)

Assertion

Ref	Expression
dibelval2nd	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (2nd ‘𝑌) = 0 )

Distinct variable groups:   𝑓,𝐾   𝑓,𝑊

Allowed substitution hints:   𝐵(𝑓)   𝑇(𝑓)   𝐻(𝑓)   𝐼(𝑓)   ≤ (𝑓)   𝑉(𝑓)   𝑋(𝑓)   𝑌(𝑓)   0 (𝑓)

Proof of Theorem dibelval2nd

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 2763	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibelval2nd.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 41918	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2849	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
10	9	biimp3a 1498	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11		xp2nd 8015	. 2 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd ‘𝑌) ∈ { 0 })
12		elsni 4606	. 2 ⊢ ((2nd ‘𝑌) ∈ { 0 } → (2nd ‘𝑌) = 0 )
13	10, 11, 12	3syl 19	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (2nd ‘𝑌) = 0 )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {csn 4589   class class class wbr 5109   ↦ cmpt 5192   I cid 5555   × cxp 5659   ↾ cres 5663  ‘cfv 6536  2^nd c2nd 7981  Basecbs 17264  lecple 17312  LHypclh 40758  LTrncltrn 40875  DIsoAcdia 41802  DIsoBcdib 41912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-2nd 7983  df-disoa 41803  df-dib 41913

This theorem is referenced by:  diblss  41944