Theorem dihglblem3aN

Description: Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihglblem.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihglblem.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihglblem.m	⊢ ∧ = ( meet ‘ 𝐾 )
	dihglblem.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
	dihglblem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihglblem.t	⊢ 𝑇 = { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) }
	dihglblem.i	⊢ 𝐽 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
	dihglblem.ih	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihglblem3aN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑇 ( 𝐼 ‘ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		dihglblem.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihglblem.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihglblem.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		dihglblem.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
5		dihglblem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		dihglblem.t	⊢ 𝑇 = { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) }
7		dihglblem.i	⊢ 𝐽 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8		dihglblem.ih	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
9	1 2 3 4 5 6	dihglblem2N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐺 ‘ 𝑆 ) = ( 𝐺 ‘ 𝑇 ) )
10	9	3adant2r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐺 ‘ 𝑆 ) = ( 𝐺 ‘ 𝑇 ) )
11	10	fveq2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ( 𝐼 ‘ ( 𝐺 ‘ 𝑇 ) ) )
12	1 2 3 4 5 6 7 8	dihglblem3N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑇 ) ) = ∩ 𝑥 ∈ 𝑇 ( 𝐼 ‘ 𝑥 ) )
13	11 12	eqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑇 ( 𝐼 ‘ 𝑥 ) )