Theorem opsrtos

Description: The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015)

	Ref	Expression
Hypotheses	opsrso.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
	opsrso.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
	opsrso.r	⊢ ( 𝜑 → 𝑅 ∈ Toset )
	opsrso.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
	opsrso.w	⊢ ( 𝜑 → 𝑇 We 𝐼 )
Assertion	opsrtos	⊢ ( 𝜑 → 𝑂 ∈ Toset )

Proof

Step	Hyp	Ref	Expression
1		opsrso.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
2		opsrso.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3		opsrso.r	⊢ ( 𝜑 → 𝑅 ∈ Toset )
4		opsrso.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
5		opsrso.w	⊢ ( 𝜑 → 𝑇 We 𝐼 )
6		eqid	⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
7		eqid	⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) )
8		eqid	⊢ ( lt ‘ 𝑅 ) = ( lt ‘ 𝑅 )
9		eqid	⊢ ( 𝑇 <bag 𝐼 ) = ( 𝑇 <bag 𝐼 )
10		eqid	⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
11		biid	⊢ ( ∃ 𝑧 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( ( 𝑥 ‘ 𝑧 ) ( lt ‘ 𝑅 ) ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( 𝑤 ( 𝑇 <bag 𝐼 ) 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) ↔ ∃ 𝑧 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( ( 𝑥 ‘ 𝑧 ) ( lt ‘ 𝑅 ) ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ( 𝑤 ( 𝑇 <bag 𝐼 ) 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) )
12		eqid	⊢ ( le ‘ 𝑂 ) = ( le ‘ 𝑂 )
13	1 2 3 4 5 6 7 8 9 10 11 12	opsrtoslem2	⊢ ( 𝜑 → 𝑂 ∈ Toset )