Theorem joinval2lem

Description: Lemma for joinval2 and joineu . (Contributed by NM, 12-Sep-2018) TODO: combine this through joineu into joinlem ?

	Ref	Expression
Hypotheses	joinval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	joinval2.l	⊢ ≤ = ( le ‘ 𝐾 )
	joinval2.j	⊢ ∨ = ( join ‘ 𝐾 )
	joinval2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
	joinval2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	joinval2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	joinval2lem	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		joinval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		joinval2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		joinval2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		joinval2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5		joinval2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		joinval2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		breq1	⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥 ) )
8		breq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑥 ↔ 𝑌 ≤ 𝑥 ) )
9	7 8	ralprg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ↔ ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ) )
10		breq1	⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧 ) )
11		breq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧 ) )
12	10 11	ralprg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 ↔ ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) ) )
13	12	imbi1d	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
14	13	ralbidv	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
15	9 14	anbi12d	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )