dibelval1st1 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st1		Structured version Visualization version GIF version

Theorem dibelval1st1 40678

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

**Hypotheses**
Ref	Expression
dibelval1st1.b	⊢ 𝐵 = (Base‘𝐾)
dibelval1st1.l	⊢ ≤ = (le‘𝐾)
dibelval1st1.h	⊢ 𝐻 = (LHyp‘𝐾)
dibelval1st1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st1.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dibelval1st1	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1^st ‘𝑌) ∈ 𝑇)

**Proof of Theorem dibelval1st1**
Step	Hyp	Ref	Expression
1		dibelval1st1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dibelval1st1.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dibelval1st1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2725	. . 3 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5		dibelval1st1.i	. . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dibelval1st 40677	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1^st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7		dibelval1st1.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	1, 2, 3, 7, 4	diael 40571	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (1^st ‘𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (1^st ‘𝑌) ∈ 𝑇)
9	6, 8	syld3an3 1406	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1^st ‘𝑌) ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6542 1^st c1st 7987 Basecbs 17177 lecple 17237 LHypclh 39512 LTrncltrn 39629 DIsoAcdia 40556 DIsoBcdib 40666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1st 7989 df-disoa 40557 df-dib 40667

This theorem is referenced by: diblss 40698

W3C validator