Theorem prdsvalstr

Description: Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 30-Apr-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Assertion	prdsvalstr	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } ) ) Struct ⟨ 1 , ; 1 5 ⟩

Proof

Step	Hyp	Ref	Expression
1		unass	⊢ ( ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ) ∪ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } ) = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } ) )
2		eqid	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ) = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } )
3	2	imasvalstr	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ) Struct ⟨ 1 , ; 1 2 ⟩
4		1nn0	⊢ 1 ∈ ℕ0
5		4nn	⊢ 4 ∈ ℕ
6	4 5	decnncl	⊢ ; 1 4 ∈ ℕ
7		homndx	⊢ ( Hom ‘ ndx ) = ; 1 4
8		4nn0	⊢ 4 ∈ ℕ0
9		5nn	⊢ 5 ∈ ℕ
10		4lt5	⊢ 4 < 5
11	4 8 9 10	declt	⊢ ; 1 4 < ; 1 5
12	4 9	decnncl	⊢ ; 1 5 ∈ ℕ
13		ccondx	⊢ ( comp ‘ ndx ) = ; 1 5
14	6 7 11 12 13	strle2	⊢ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } Struct ⟨ ; 1 4 , ; 1 5 ⟩
15		2nn0	⊢ 2 ∈ ℕ0
16		2lt4	⊢ 2 < 4
17	4 15 5 16	declt	⊢ ; 1 2 < ; 1 4
18	3 14 17	strleun	⊢ ( ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ) ∪ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } ) Struct ⟨ 1 , ; 1 5 ⟩
19	1 18	eqbrtrri	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , 𝑂 ⟩ , ⟨ ( le ‘ ndx ) , 𝐿 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , ∙ ⟩ } ) ) Struct ⟨ 1 , ; 1 5 ⟩