Theorem dibelval1st2N 41154

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 2736	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5		dibelval1st2.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dibelval1st 41152	. 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 41047	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋)
10	6, 9	syld3an3 1410	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5142  ‘cfv 6560  1^st c1st 8013  Basecbs 17248  lecple 17305  LHypclh 39987  LTrncltrn 40104  trLctrl 40161  DIsoAcdia 41031  DIsoBcdib 41141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-1st 8015  df-disoa 41032  df-dib 41142

This theorem is referenced by: (None)