Theorem dibelval1st2N 42188

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

Hypotheses

Ref	Expression
dibelval1st2.b	⊢ 𝐵 = (Base‘𝐾)
dibelval1st2.l	⊢ ≤ = (le‘𝐾)
dibelval1st2.h	⊢ 𝐻 = (LHyp‘𝐾)
dibelval1st2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st2.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dibelval1st2.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibelval1st2N	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋)

Proof of Theorem dibelval1st2N

Step	Hyp	Ref	Expression
1		dibelval1st2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dibelval1st2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dibelval1st2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2761	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5		dibelval1st2.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dibelval1st 42186	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7		dibelval1st2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		dibelval1st2.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9	1, 2, 3, 7, 8, 4	diatrl 42081	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋)
10	6, 9	syld3an3 1436	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  ‘cfv 6537  1^st c1st 7997  Basecbs 17380  lecple 17428  LHypclh 41021  LTrncltrn 41138  trLctrl 41195  DIsoAcdia 42065  DIsoBcdib 42175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7999  df-disoa 42066  df-dib 42176

This theorem is used by: (None)