Theorem dibelval2nd 41644

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 2739	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibelval2nd.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 41636	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2825	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
10	9	biimp3a 1477	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11		xp2nd 7964	. 2 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd ‘𝑌) ∈ { 0 })
12		elsni 4572	. 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 )

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {csn 4555   class class class wbr 5072   ↦ cmpt 5153   I cid 5512   × cxp 5616   ↾ cres 5620  ‘cfv 6485  2^nd c2nd 7930  Basecbs 17170  lecple 17218  LHypclh 40476  LTrncltrn 40593  DIsoAcdia 41520  DIsoBcdib 41630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-2nd 7932  df-disoa 41521  df-dib 41631

This theorem is referenced by:  diblss  41662