Theorem ply1sclf1

Description: The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
	ply1sclid.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	ply1sclf1.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
Assertion	ply1sclf1	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 –1-1→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
3		ply1sclid.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		ply1sclf1.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
5	1 2 3 4	ply1sclf	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 ⟶ 𝐵 )
6		fveq2	⊢ ( ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑦 ) → ( coe1 ‘ ( 𝐴 ‘ 𝑥 ) ) = ( coe1 ‘ ( 𝐴 ‘ 𝑦 ) ) )
7	6	fveq1d	⊢ ( ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑦 ) → ( ( coe1 ‘ ( 𝐴 ‘ 𝑥 ) ) ‘ 0 ) = ( ( coe1 ‘ ( 𝐴 ‘ 𝑦 ) ) ‘ 0 ) )
8	1 2 3	ply1sclid	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾 ) → 𝑥 = ( ( coe1 ‘ ( 𝐴 ‘ 𝑥 ) ) ‘ 0 ) )
9	8	adantrr	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → 𝑥 = ( ( coe1 ‘ ( 𝐴 ‘ 𝑥 ) ) ‘ 0 ) )
10	1 2 3	ply1sclid	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾 ) → 𝑦 = ( ( coe1 ‘ ( 𝐴 ‘ 𝑦 ) ) ‘ 0 ) )
11	10	adantrl	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → 𝑦 = ( ( coe1 ‘ ( 𝐴 ‘ 𝑦 ) ) ‘ 0 ) )
12	9 11	eqeq12d	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → ( 𝑥 = 𝑦 ↔ ( ( coe1 ‘ ( 𝐴 ‘ 𝑥 ) ) ‘ 0 ) = ( ( coe1 ‘ ( 𝐴 ‘ 𝑦 ) ) ‘ 0 ) ) )
13	7 12	syl5ibr	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ) ) → ( ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑦 ) → 𝑥 = 𝑦 ) )
14	13	ralrimivva	⊢ ( 𝑅 ∈ Ring → ∀ 𝑥 ∈ 𝐾 ∀ 𝑦 ∈ 𝐾 ( ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑦 ) → 𝑥 = 𝑦 ) )
15		dff13	⊢ ( 𝐴 : 𝐾 –1-1→ 𝐵 ↔ ( 𝐴 : 𝐾 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐾 ∀ 𝑦 ∈ 𝐾 ( ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
16	5 14 15	sylanbrc	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 –1-1→ 𝐵 )