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Metamath Proof Explorer

Theorem xpsless

Description: Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015)

		Ref	Expression
	Hypotheses	xpsle.t	⊢ 𝑇 = ( 𝑅 ×_s 𝑆 )
		xpsle.x	⊢ 𝑋 = ( Base ‘ 𝑅 )
		xpsle.y	⊢ 𝑌 = ( Base ‘ 𝑆 )
		xpsle.1	⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
		xpsle.2	⊢ ( 𝜑 → 𝑆 ∈ 𝑊 )
		xpsle.p	⊢ ≤ = ( le ‘ 𝑇 )
	Assertion	xpsless	⊢ ( 𝜑 → ≤ ⊆ ( ( 𝑋 × 𝑌 ) × ( 𝑋 × 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xpsle.t	⊢ 𝑇 = ( 𝑅 ×_s 𝑆 )
2		xpsle.x	⊢ 𝑋 = ( Base ‘ 𝑅 )
3		xpsle.y	⊢ 𝑌 = ( Base ‘ 𝑆 )
4		xpsle.1	⊢ ( 𝜑 → 𝑅 ∈ 𝑉 )
5		xpsle.2	⊢ ( 𝜑 → 𝑆 ∈ 𝑊 )
6		xpsle.p	⊢ ≤ = ( le ‘ 𝑇 )
7		eqid	⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } )
8		eqid	⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 )
9		eqid	⊢ ( ( Scalar ‘ 𝑅 ) X_s { ⟨ ∅ , 𝑅 ⟩ , ⟨ 1_o , 𝑆 ⟩ } ) = ( ( Scalar ‘ 𝑅 ) X_s { ⟨ ∅ , 𝑅 ⟩ , ⟨ 1_o , 𝑆 ⟩ } )
10	1 2 3 4 5 7 8 9	xpsval	⊢ ( 𝜑 → 𝑇 = ( ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) “_s ( ( Scalar ‘ 𝑅 ) X_s { ⟨ ∅ , 𝑅 ⟩ , ⟨ 1_o , 𝑆 ⟩ } ) ) )
11	1 2 3 4 5 7 8 9	xpsrnbas	⊢ ( 𝜑 → ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) = ( Base ‘ ( ( Scalar ‘ 𝑅 ) X_s { ⟨ ∅ , 𝑅 ⟩ , ⟨ 1_o , 𝑆 ⟩ } ) ) )
12	7	xpsff1o2	⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ( 𝑋 × 𝑌 ) –1-1-onto→ ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } )
13		f1ocnv	⊢ ( ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ( 𝑋 × 𝑌 ) –1-1-onto→ ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) → ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) –1-1-onto→ ( 𝑋 × 𝑌 ) )
14	12 13	mp1i	⊢ ( 𝜑 → ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) –1-1-onto→ ( 𝑋 × 𝑌 ) )
15		f1ofo	⊢ ( ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) –1-1-onto→ ( 𝑋 × 𝑌 ) → ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) –onto→ ( 𝑋 × 𝑌 ) )
16	14 15	syl	⊢ ( 𝜑 → ^◡ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) : ran ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ { ⟨ ∅ , 𝑥 ⟩ , ⟨ 1_o , 𝑦 ⟩ } ) –onto→ ( 𝑋 × 𝑌 ) )
17		ovexd	⊢ ( 𝜑 → ( ( Scalar ‘ 𝑅 ) X_s { ⟨ ∅ , 𝑅 ⟩ , ⟨ 1_o , 𝑆 ⟩ } ) ∈ V )
18	10 11 16 17 6	imasless	⊢ ( 𝜑 → ≤ ⊆ ( ( 𝑋 × 𝑌 ) × ( 𝑋 × 𝑌 ) ) )