Theorem dibelval2nd 41737

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 2761	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7		dibelval2nd.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dibval2 41729	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
9	8	eleq2d 2847	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
10	9	biimp3a 1489	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11		xp2nd 7998	. 2 ⊢ (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd ‘𝑌) ∈ { 0 })
12		elsni 4596	. 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {csn 4579   class class class wbr 5097   ↦ cmpt 5178   I cid 5537   × cxp 5641   ↾ cres 5645  ‘cfv 6516  2^nd c2nd 7964  Basecbs 17236  lecple 17284  LHypclh 40569  LTrncltrn 40686  DIsoAcdia 41613  DIsoBcdib 41723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-2nd 7966  df-disoa 41614  df-dib 41724

This theorem is referenced by:  diblss  41755